FFERAL CURVES AND MINIMAL SETS
JOEL W. FISH AND HELMUT HOFER
Abstract.
Here we prove that for each Hamiltonian function H ∈ C ∞ ( R , R )defined on the standard symplectic ( R , ω ), for which M := H − (0) is a non-empty compact regular energy level, the Hamiltonian flow on M is not minimal.That is, we prove there exists a closed invariant subset of the Hamiltonian flowin M that is neither ∅ nor all of M . This answers the four dimensional case ofa twenty year old question of Michel Herman, part of which can be regardedas a special case of the Gottschalk Conjecture.Our principal technique is the introduction and development of a new classof pseudoholomorphic curve in the “symplectization” R × M of framed Hamil-tonian manifolds ( M, λ, ω ). We call these feral curves because they are allowedto have infinite (so-called) Hofer energy, and hence may limit to invariant setsmore general than the finite union of periodic orbits. Standard pseudoholo-morphic curve analysis is inapplicable without energy bounds, and thus muchof this manuscript is devoted to establishing properties of feral curves, suchas area and curvature estimates, energy thresholds, compactness, asymptoticproperties, etc.
Contents
1. Introduction and Results 21.1. Context: Dynamical Systems 41.2. Context: Pseudoholomorphic Curves 61.3. Overview of Results 112. Background 172.1. Ambient Geometric Structures 182.2. Pseudoholomorphic Curves 293. Existence of Minimal Subsets 383.1. Proof of Theorem 1 393.2. Proof of Theorem 2 524. Supporting Proofs 544.1. Proof of Theorem 3: Exponential Area Bounds 544.2. Proof of Theorem 4: ω -Energy Threshold 794.3. Proof of Theorem 5: Asymptotic Connected-Local Area Bound 814.4. Proof of Theorem 6: Asymptotic Curvature Bound 1234.5. Proof of Theorem 7: Existence Workhorse 133Appendix A. Minor Miscellanea 158References 166 Key words and phrases.
Gottschalk, Herman, Hamiltonian, feral, minimal set.The first author’s research in development of this manuscript was supported in part by theEllentuck Fund, the Fund for Math at the Institute for Advanced Study, and NSF-DMS StandardResearch Grant Award 1610453 . a r X i v : . [ m a t h . S G ] D ec J.W. FISH AND H. HOFER Introduction and Results
Almost since their inception, pseudoholomorphic curves have been the commonthread by which symplectic geometry, topology, and Hamiltonian dynamics havebeen intertwined. Specifically, these curves generalize the notion of holomorphiccurves in a complex manifold to curves in an almost complex symplectic manifold;moreover, they do so while preserving a variety of robust properties which detectsubtle geometric aspects, dynamical features, and relationships between the two. Atits core, this manuscript is about the discovery of a new class of pseudoholomorphiccurve (with reasonable properties) and their application toward answering a twentyyear old question of Michel Herman [21] raised at the 1998 ICM. We should mention,this new class of potentially infinite energy curve seems to have been very difficultto predict, particularly as a natural extension of finite energy curves. We elaboratefurther on this in Section 1.3. What follows are two separate but inextricably linkedresults, each of which is of notable interest to a separate camp of mathematician:the dynamicist and symplectic topologist. We begin with an easy to state variant ofour main dynamical theorem and provide some brief discussion of the significanceof the result and its proof.
Theorem 1. (Main dynamical result)Consider R equipped with the standard symplectic structure and a Hamiltonian H ∈ C ∞ ( R , R ) for which M := H − (0) is a non-empty compact regular energylevel. Then the Hamiltonian flow on M is not minimal. Recall that a flow is minimal provided that every trajectory is dense; or, in otherwords, if there exist no closed invariant subsets other than the empty set and thetotal space. Specialists in dynamical systems may regard the above result as aproof of a Hamiltonian version of the Gottschalk conjecture; for additional details,see Section 1.1 below. For symplectic topologists, the dynamical result itself isperhaps less important than the proof, which heavily uses a new class of pseudo-holomorphic curve, so called feral curves . This new class of pseudoholomorphiccurves opens the door for further studies of symplectic cobordisms without theusual requirement that the boundaries are of contact or stable Hamiltonian type.For the current theory of pseudoholomorphic curves, such requirements have beentechnical necessities.To sketch the proof idea of Theorem 1, we note that the key technique weemploy is both venerably old and radically new: Given our smooth hypersurface M := H − (0) ⊂ R , we symplectically embed a neighborhood of M into C P ,stretch the neck along this hypersurface, use Gromov’s existence result for degreeone pseudoholomorphic spheres, show these curves stretch as they fall into thenegative symplectization end, and then establish a compactness result which yieldsa non-compact pseudoholomorphic curve in R × M which limits to the desired closedinvariant subset.The novelty here is not so much the simple geometric idea underlying the proof,but rather that the proof can be made to work at all. Specifically, because thehypersurface H − (0) is neither contact type nor stable-Hamiltonian type, we donot have a priori Hofer energy bounds as we stretch the neck. The result of ouranalysis is then to find a potentially infinite energy pseudoholomorphic curve, whichhas surprisingly nice properties. For example, the ends detect the desired closedinvariant subset. In either case, these curves have the interesting feature that each ERAL CURVES AND MINIMAL SETS 3 can interpolate between finite and infinite energy ends, and in families these curvescan interpolate between finite and infinite energy curves. Precisely because of thisability to transition between the tame (finite energy) and the wild (infinite energy),we have picked the name feral curves .Those familiar with pseudoholomorphic curves in symplectizations should readilybe aware of the fact that finiteness of, and a priori bounds on, Hofer energy isan absolute bedrock assumption upon which a tremendous number of additionalproperties are built. By removing this assumption, we must return to basics, andit should be no surprise then that this takes considerable effort. In particular thewidely used domain-centric approach of predominantly regarding curves as maps,must be replaced by a more target-centric approach which treats curves more likesubmanifolds. The origin of this more target-centric approach was likely Taubes’work in [35] which regarded curves as integral currents, however the techniquestherein are too coarse for our needs here. Instead we build on a mixture of ideasinitiated in the alternate approach to compactness in Symplectic Field Theory provided by Kai Cieliebak and Klaus Mohnke in [6], and then heavily generalizedby the first author of this manuscript in [12].In Section 1.2 below, we elaborate on the difficulties involved with analyzingcurves of infinite energy, but note that the end result is the establishment of adecidedly novel class of pseudoholomorphic curve equipped with many propertieswhich are not dissimilar from finite energy curves, and moreover which stronglysuggest a rich avenue of future research. Indeed, one natural direction would be toexplore whether there exists a homology theory akin to ECH , or SFT but which hasgenerators which are dynamical structures other than (weighted) sets of periodicorbits. An alternate direction would be to explore the possibility that for a generic framework , feral curves actually have finite energy, and hence Symplectic FieldTheory has extension to symplectic manifolds with generic boundary rather thancontact-type or stable Hamiltonian boundary.At present we outline the remainder of the manuscript. First, in Section 1.1, weprovide some historical context for Theorem 1 from the perspective of dynamicalsystems. In Section 1.2, we elaborate on the historical context from the pseudo-holomorphic curve perspective, and we highlight some of the potential difficultiesthat must be resolved in order to prove Theorem 1. We finish this introduction withSection 1.3 which provides an overview of the additional theorems proved in thismanuscript. Then, in Section 2, we provide background definitions and state someknown results which will be used throughout later proofs. Most of the material inthis section is likely to be familiar to those comfortable with pseudoholomorphiccurve analysis, however there are a number of definitions which may be novel. InSection 3 we provide the main argument which establishes Theorem 1. This proofrelies on several technical supporting results, and these are restated and proved inSection 4. Symplectic Field Theory was introduced by Eliashberg, Givental, and Hofer in [9], and morerecently a very nice overview and background was provided by Chris Wendl in [39]. For a nice introduction to Embedded Contact Homology, see [25], and for some nice dynamicalapplications thereof, see Daniel Cristofaro-Gardiner, Michael Hutchings, and Vinicius Ramos in[7], as well as Masayuki Asaoka and Kei Irie in [2]. Perhaps using abstract perturbations.
J.W. FISH AND H. HOFER
Context: Dynamical Systems.
In the 1950s, the following two importantconjectures about autonomous flows on S were stated: Seifert Conjecture:
Every non-singular flow on S has a periodic orbit. Gottschalk Conjecture: S does not support a minimal flow. Regarding the history of these two conjectures, we begin with the Seifert Con-jecture. After being posed in the early 1950s, it stood as an open problem for overtwenty years, until 1974 when Paul Schweitzer [32] proved the existence of a C vector field on S with no closed orbits. The existence of such a vector field thendisproved the Seifert Conjecture, and hence Schweitzer’s vector field was regardedas a C counterexample. Of course, C vector fields are of rather low regularity,and hence comprise a rather broad class of vector fields, so it is natural to ask ifthere are more restrictive classes of vector fields, say of higher regularity, for whichthe Seifert Conjecture is true. And indeed, over the next thirty years, this questionwas raised and answered in the negative for flows of increasing regularity. For ex-ample, in 1988 Jenny Harrison [20] adapted Schweitzer’s argument to find a C δ counterexample. Using very different techniques, in 1994 Krystyna Kuperberg [28]found a C ∞ smooth counterexample to the Seifert Conjecture, and in 1996 GregKuperberg and Krystyna Kuperberg [27] established an analytic counterexample.Also in 1996, Greg Kuperberg [26] found a volume preserving C counterexampleto the Seifert conjecture.With so many counterexamples established, the Seifert Conjecture seemed defini-tively disproved, with one notable exception: Reeb flows. Indeed, in 1993 HelmutHofer [23] proved that every C ∞ Reeb vector field on S generates a periodic orbit.A corollary of this result is the following. Let Ω be a smooth volume form on S and X a nonsingular volume preserving vector field. Then it holds d ( i X Ω) = 0 and since H ( S , R ) = 0 we can find a 1-form λ satisfying dλ = i X Ω. Since H ( S , R ) = 0 anyprimitive λ (cid:48) of i X Ω differs from λ by the differential of a smooth map h : S → R ,i.e. λ (cid:48) = λ + dh . Hofer’s theorem implies that in the case where a primitive λ (cid:48) of i X Ω can be found satisfying λ (cid:48) ( X ( x ))) (cid:54) = 0 for all x ∈ S , there exists a periodicorbit.There are several points of note regarding Hofer’s 1993 result. Of particular in-terest is how heavily it relied on deep results from contact topology, like Eliashberg’sclassification of overtwisted contact three-manifolds as either tight or overtwisted,see [8]; Bennequin’s proof that the standard contact structure on S is tight, see[3]; and Eliashberg’s complete classification of contact S , see [10]. To establishexistence of periodic Reeb orbits, Hofer built on the theory of pseudoholomorphiccurves introduced by Gromov in [19], and on Floer’s idea to use them to find peri-odic orbits of Hamiltonian vector fields as in [14]. It is worth noting that Hofer’stechniques were quite robust, and although not explicitly used to do so in [23], theywere capable of recovering Rabinowitz’s results in [31] which guarantee the exis-tence of periodic Reeb orbits on the boundary of star-shaped domains in R . Hofer’sapproach is relevant, since it is the principle idea behind the proof of Theorem 1above.Before proceeding, it is important to highlight a result which should be kept inmind, and held in contrast to Theorem 1 namely: ERAL CURVES AND MINIMAL SETS 5
Theorem 1.1 (2003, Ginzburg-G¨urel [17]) . There exists a proper C -smooth function H : R → R , for which H − (0) (cid:39) S is aregular level set on which the Hamiltonian flow has no periodic orbits. The focus here should not be on the relatively low regularity of the Hamiltonian,but rather on the non-existence of a periodic orbit. Indeed, the relevance is thatwhile the above result guarantees non-existence of any periodic orbits, the principleresult of this manuscript guarantees the existence of a closed flow-invariant propersubset as a type of limit set of a pseudoholomorphic curve; this is discussed furtherin Section 1.2 below. In particular then, this suggests that the new class of curvesexplored below indeed find closed invariant subsets more general than periodicorbits. We note one slight caveat: Our analysis here is done in regularity C ∞ , whilethe Ginzburg-G¨urel result holds in C . Nevertheless we believe both results can begeneralized to reach the desired conclusion. Indeed, the constructions in the presentpaper should be doable in a C α -frame work. Also, in [17], the authors remark: “It is quite likely that our construction gives an embedding S → R without closedcharacteristics, which is C α -smooth.”With these results established, the answer to the Seifert Conjecture is well un-derstood and essentially complete: It is false for vector fields as regular as onelikes, and false for volume preserving flows, but true for Reeb flows. At this pointwe turn our attention to Question 2 and the Gottschalk Conjecture, and we beginby noting that the lack of progress on this problem stands in stark contrast tothe nearly complete understanding of the Seifert Conjecture. Indeed, despite morethan a half century worth of attempts, no essential progress has been made on theGottschalk Conjecture. We make two important qualifications to that statement.First, strictly speaking, results stated above which guarantee existence of periodicorbits, for example [23] and [31], are progress on the Gottschalk Conjecture for theclass of Reeb vector fields, however because the closed invariant sets are always pe-riodic orbits, this is more a result about the Seifert Conjecture than the GottschalkConjecture. Second, although there has been no direct progress on the GottschalkConjecture, there have been a variety of results on related problems. For exam-ple, in 2009 Clifford Taubes [36] proved that a volume preserving vector field ona compact 3-manifold whose dual 2-form is exact (such as S ) can not generateuniquely ergodic dynamics unless its asymptotic linking number is zero; in 2014Bassam Fayad and Anatole Katok [11] construct analytic uniquely ergodic (henceminimal) volume preserving maps (but not flows) on odd dimensional spheres; andin 2015 Ginzburg and Niche [18] showed that the autonomous Hamiltonian flow ona compact regular energy level in R n (and somewhat more generally) cannot beuniquely ergodic.In short, results in the direction of the Gottschalk Conjecture have been one oftwo types, namely either establishing the existence of periodic orbits as in the Reebcase, or else making definitive progress on a related problem. As such, we notethat it is somewhat surprising that more direct progress has not been made giventhe importance of this problem. For example, if the Gottschalk conjecture is true,then in all likelihood a method to prove it will need to develop a global theory forfinding closed invariant subsets, which in turn will touch on long-standing questionsin dynamical systems, particularly in cases in which flows are volume-preserving. Itis also worth noting that Gottschalk’s question has been well established as histor-ically significant. Indeed, it was raised in 1974 during the American Mathematical J.W. FISH AND H. HOFER
Society’s special symposium on the mathematical consequences of Hilbert’s prob-lems [5]. It made another appearance in [34] when mentioned by Steven Smale inhis list of the most important problems for the twenty-first century. And it ap-peared again in 1998 at the International Congress of Mathematics during MichaelHerman’s talk [21], in which he raised the following related question.
Question: (1998, Herman)
When n ≥ , can one find a C ∞ compact, connected,regular hypersurface in R n on which the characteristic flow is minimal? Recall that the characteristic flow is just the Hamiltonian flow associated to anysmooth Hamiltonian for which the hypersurface is a regular energy level. Con-sequently, Herman’s question might be regarded as the Hamiltonian Gottschalkconjecture for compact energy levels in R n , and the principle result of this man-uscript is to answer his question in the negative when n = 2, i.e. a version of theGottschalk conjecture holds for compact regular Hamiltonian energy surfaces in R . We complete this section by stating a conjecture, which seems plausible giventhe developments in this paper. It combines a question about almost existence ofperiodic orbits, a well-studied problem, with the existence question of proper closedinvariant subsets. Conjecture 1 (minimal sets in energy piles) . Assume that Ω is a symplectic form on [ − , × S and denote by H : [ − , × S → R the Hamiltonian defined by H ( t, m ) = t . Denote by Σ t the regular compact energysurface H − ( t ) and define the subset S ⊂ [ − , to consist of all t for which theenergy surface Σ t carries a periodic orbit. Then the following holds:(1) measure ( S ) = 2 (2) For t ∈ [ − , \ S there exists a closed proper invariant subset for theHamiltonian flow on Σ t . Context: Pseudoholomorphic Curves.
In 1985 Mikhail Gromov [19] in-troduced the notion of pseudoholomorphic curves in almost complex manifolds.Such curves were a generalization from holomorphic curves in complex manifolds,to curves in real manifolds equipped by a preferred rotation by 90 degrees in thetangent bundle (determined by an almost complex structure; see Definition 2.2below). Roughly speaking then, a pseudoholomorphic curve is a map from a Rie-mann surface into a manifold equipped with an almost complex structure with theproperty that the derivative of the map intertwines the complex structure on theRiemann surface with the almost complex manifold on the target.These curves solve an elliptic partial differential equation and they form the zeroset of a non-linear Fredholm operator and thus tend to live in smooth families.A crucial observation by Gromov was that if the almost complex structure J istamed by a symplectic form, then curves in a fixed homology class will have apriori bounded energy and area, and hence they degenerate in a manner which isessentially indistinguishable from the manner in which algebraic curves degeneratein smooth projective varieties; from a geometric analysis perspective, this is alsoessentially the same manner in which minimal surfaces degenerate in Riemannianmanifolds. Put another way, modulo the formation of nodal or cusp curves, familiesof pseudoholomorphic curves of a fixed homology class are compact; this is thecelebrated Gromov compactness theorem for pseudoholomorphic curves. Moreover,algebraic counts of curves have yielded the so-called Gromov-Witten invariants. ERAL CURVES AND MINIMAL SETS 7
In 1986, shortly after Gromov’s seminal paper, Andreas Floer [14] discoveredthat an inhomogeneous version of the pseudoholomorphic curve equation could beused to study the Morse homology of the loop space of a closed symplectic manifold.Here the Morse function was the symplectic action functional associated to a one-periodic Hamiltonian function. In turn, this action functional had one-periodicorbits of a Hamiltonian flow as critical points, and with such orbits as generators,the differential was determined by counting perturbed pseudoholomorphic cylinders(the so called Floer trajectories) between such orbits. The resulting theory hasbecome known as Hamiltonian Floer homology.Then in 1993, Helmut Hofer [23] considered a sort of hybrid case: pseudoholomor-phic curves in symplectizations of contact manifolds. Here the interesting featurewas that the curves had infinite area, but had finite Hofer-energy; or equivalently,uniformly bounded local-area. It turned out that such curves were asymptotic tocylinders over periodic Reeb orbits. Furthermore, these curves were either pos-itively or negatively asymptotic to such orbit cylinders, and hence under certainhypotheses one could construct a variety of flavors of contact homology (cylindrical,linearized, full, rational, embedded, etc), in which the generators are certain setsof (sometimes weighted) periodic Reeb orbits, and with the differential determinedby counting certain finite energy pseudoholomorphic curves which positively limitto one orbit set and negatively limit to another orbit set.It was eventually discovered that each of these theories (Gromov-Witten invari-ants, Hamiltonian Floer homology, contact homology, etc) was subsumed in a largerSymplectic Field Theory (SFT) proposed by Eliashberg, Givental, and Hofer in [9].More precisely, the moduli spaces of pseudoholomorphic curves that generate eachof these theories is contained in the collection of moduli spaces studied in SFT.An absolutely crucial feature in each of these theories is that the curves in ques-tion have an a priori energy bound, which should be deduced from representing afixed homology class, and which in turn guarantees a (local) area bound. Indeed,without such energy control, pseudoholomorphic curves have notably wild behav-ior. For example, in the symplectization of a contact manifold, R × M , for anyadmissible almost complex structure and any Reeb trajectory γ : R → M , the map( s, t ) (cid:55)→ ( s, γ ( t )) ∈ R × M is pseudoholomorphic and of infinite energy and whichmay have an image which is dense in R × M ; we call such curves pseudoholomorphic sheets . As a consequence of the apparent wild behavior of infinite energy curves,both popular and expert belief has been that there exists a dichotomy among pseu-doholomorphic curves: those with energy bounds and those without. Moreover, theformer are tame and well understood while the latter have such wild behavior thatone cannot feasibly hope study them in a meaningful way.To illustrate this idea, we draw an analogy with holomorphic functions on thepunctured complex plane. Here, of course, there is a dichotomy, namely functionswith poles versus functions with essential singularities. The former are meromorphicfunctions and are algebraic in nature, while the latter are especially unmanageable,particularly in light of Picard’s Great Theorem, which states that in each neigh-borhood of an essential singularity, a holomorphic function takes on every complexvalue (except possibly one) infinitely many times. This clear division of holomor-phic functions has long been assumed to carry over into the realm of SymplecticField Theory: curves either have bounded energy, are tame, and are well under-stood, or else they have unbounded energy, are wild, and are unmanageable. One of J.W. FISH AND H. HOFER the main thrusts of this manuscript is to defy conventional wisdom, and illuminatean intermediate class of infinite energy curves. Or, perhaps more accurately, iden-tify a class of curves which appears to interpolate between tame and wild curves,which we designate as feral curves. We give a precise formulation of feral curvesin Definition 1.5 below, but roughly speaking they are proper pseudoholomorphicmaps u : S → R × M into symplectizations of framed Hamiltonian manifolds forwhich S has finite topology (genus, connected components, etc.) and (cid:82) S u ∗ ω < ∞ .We note that on one hand, the properness condition rules out the aforementionedpseudoholomorphic sheets ( s, t ) (cid:55)→ ( s, γ ( t )), and the finite ω -energy condition tendsto prevent such curves becoming too wild, however, by not requiring the Hamil-tonian structure to be stable allows feral curves to have infinite Hofer energy, andindeed we expect that some definitely do.Before proceeding, we aim to give some idea of how difficult it is to study pseu-doholomorphic curves without a priori bounded energy, so we take a moment tostep through some potential issues. As a model starting point, one might considera sequence of finite energy planes, all asymptotic to the same simply covered orbitcylinder, and study what might happen as one progresses through the sequencewhile assuming the Hofer energy tends to infinity. First, the SFT compactnesstheorem for pseudoholomorphic curves [4] does not apply directly, since energy isunbounded. Nevertheless, one might mimic the argument to see where it breaksdown. In this model case, the conformal structures on the domain Riemann sur-faces do not change, so the key issue is whether or not the gradient is bounded; ifboundedness fails, we attempt bubbling analysis. This is where difficulties start toarise.In Gromov-Witten theory, if the gradient blows up, then rescaling analysis ex-tracts a sphere-bubble, which captures a threshold amount of energy. In SFTcompactness something similar occurs, except that rescaling analysis extracts ei-ther a sphere-bubble or else a finite energy plane, and either object captures athreshold amount of ω/dλ -energy, so the process terminates after finitely manyiterations. But without energy bounds, we cannot guarantee that a finite energyplane bubbles off – instead one might only be able to extract something akin to aninfinite energy sheet, which has arbitrarily small ω/dλ -energy. Worse still, withoutsome threshold amount of energy being captured via rescaling analysis, one canno longer guarantee that the gradient blows up only in a neighborhood of finitelymany points. Indeed, a priori the gradient could blow up everywhere.Still, maybe by some alternate methods, or by considering a model example, onecould perhaps extract something like an infinite energy plane which has finite ω -energy. However, even in such a case, two possibilities complicate matters further.First, a priori, it need not be the case that the domain Riemann surface of sucha curve is conformally equivalent to the complex plane; it could be an open diskinstead. In the SFT setting, it is usually assumed that the domains of curves areconformally equivalent to punctured Riemann surfaces, however this is an assump-tion which can be removed and then easily deduced from other standard analysis.However, for infinite energy curves it is a possibility which needs to be more seri-ously considered. Second, given a single proper infinite energy plane (or disk, as thecase may be), it need not be the case that the gradient is globally bounded. Again, For a precise formulation of a framed Hamiltonian manifold see Definition 2.4 below, howeverat present we note that it is more general than both contact and stable Hamiltonian.
ERAL CURVES AND MINIMAL SETS 9 in the usual SFT setting, this can be deduced in a variety of ways which dependon asymptotic analysis or finiteness of energy, but in the infinite energy case it is apossibility that must again be considered.To summarize the difficulties, we see that once we remove a priori energy bounds,SFT compactness does not apply, there is no local area bound, there is no energythreshold, there is potentially dense gradient blowup, a single curve can have un-bounded gradient, and even something simple like an infinite energy “plane” mightin fact be holomorphically parametrized by an open disk, or its image may bedense in the target manifold. In short, without energy bounds our arsenal of stan-dard pseudoholomorphic techniques becomes largely ineffectual, and curve analysisrapidly appears unmanageable. Those somewhat familiar with pseudoholomorphiccurves can then perhaps see the difficulty faced at the outset: With so many ba-sic tools rendered inapplicable, it becomes exceedingly difficult to formulate whatproperties to expect, let alone prove them.Nevertheless, despite these obstacles, analysis is still possible, and it should notbe surprising that a bulk of this manuscript is dedicated to establishing sufficientproperties to prove the main dynamical result. An overview of these results isprovided in Section 1.3 below, but at present we provide an alternate characteriza-tion of feral curves which may be less amenable to analysis but which is better forproviding a conceptual framework.To that end, we first back up and re-characterize finite energy curves insidesymplectizations of contact manifolds, where ω = dλ , as follows. Outside a largecompact set, say [ − n, n ] × M for n (cid:29)
1, a finite Hofer-energy curve is immersed,and the tangent planes are nearly vertical; that is, they are nearly tangent to thetwo-plane distribution ker ω ⊂ T ( R × M ). Consequently, outside a large compactset, one can project the asymptotic ends of a curve into the manifold M andregard this as a path of loops parameterized by level sets of the symplectizationcoordinate R . Of interest here is the fact that such a path of loops is in factan integral curve of a gradient-like vector field on the loop space of M which hasperiodic Reeb orbits as rest points. Keeping this in mind, one can then regard finiteenergy pseudoholomorphic curves as submanifolds which can be geometrically ortopologically interesting in some large compact sets of R × M , like inside [ − n, n ] × M ,but outside of this compact set they can morally be thought of as gradient flow linesconverging to critical points of a functional on the loop space of M . The surprisingfeature of feral curves is that they can be thought of in nearly the same way. Indeed,as we make clear below, outside a large compact set a feral pseudoholomorphic curveis immersed with tangent planes nearly vertical. Again, the result is that the endsof a feral curve can be regarded as path of loops in M , and this path is in factan integral curve of a gradient-like vector field. The key difference however, whichstands in stark contrast with the contact and stable Hamiltonian case, is that in thegeneral framed Hamiltonian case the action functional is not Palais-Smale. Morespecifically, feral curves have ends which are “gradient” flow lines along which theaction is bounded but the trajectory escapes to infinity.The above characterization of feral curves is then both a boon and a curse.The upside is that despite the fact that curves without energy bounds seem wildlyunmanageable, we show that feral curves nevertheless have a surprising numberof properties which make their study tractable and somewhat familiar, if non-standard. Moreover, feral curves still lie in the general heuristic framework in which pseudoholomorphic curves are of type of generalized gradient flow line, andhence could be used to define some generalized version of Morse homology or a morecomplicated algebraic invariant like Symplectic Field Theory. The great downsidethough, is that Morse theory for a general non-Palais-Smale functional is an illconceived notion, and at best it is unlikely to be an invariant, and at worst itsimply cannot be defined. Indeed, in some sense, the general action functional inthe framed Hamiltonian case appears to have “critical points at infinity,” which,at present, defy direct analysis, and hence preclude a complete SFT compactnesstheorem for feral curves, as well as a Fredholm theory, a gluing theory, and areasonable hope of an algebraic invariant.It is possible that the above characterization, and the potential problems itbrings, may give the impression of casting a dark shadow over the landscape ofpossibilities for feral curves. We take a moment then to highlight certain glimmersof hope. First, we note that in examples, feral curves tend to have ends with arather nice property: They tend to limit to a finite collection of hyperbolic minimalsets connected by families of heteroclinic trajectories. Or, more geometrically then,while we have become used to pseudoholomorphic curves bubbling or breaking (asin Floer homology, contact homology, etc) and limiting to periodic orbits, now itseems possible that periodic orbits can themselves bubble or break and that feralcurves detect this and limit to the broken orbit. This raises a question: If one cananalytically understand the violent breaking and gluing phenomena in Morse-likehomology theories, then why can one not adapt the analysis to understand curveslimiting to broken periodic orbits as well? Perhaps one can. Or perhaps one mustregularize the space of periodic orbits, broken or not, in a fashion similar to reg-ularizing moduli spaces of pseudoholomorphic curves before defining a differentialor more complicated algebraic invariant. In either case, these possibilities warrantinvestigation.Finally, we raise an important, and perhaps deeper, question. Question:
Are feral pseudoholomorphic curves essential or inessential?
We elaborate. One perspective is that rather fundamentally, pseudoholomor-phic curves detect topology of a symplectic nature. Thus when pseudoholomorphiccurves behave unexpectedly, there are roughly two possibilities. The first is that theodd behavior is somehow non-generic and therefore is likely to be inconsequential.The second is that the pseudoholomorphic curves in question are actually detectingan unexpected topological feature, and thus such curves, and the detected phenom-ena, are important and essential. It is hopefully clear that answering the abovequestion – in either direction – is an important avenue of research.To close this section, we bring the discussion back, almost full circle, to thetame/wild dichotomy, and how feral curves fit comfortably in neither class, butrather share properties of each. A consequence is that they provide a definitive op-portunity to push pseudoholomorphic curves beyond their conventional limitationsand possibly discover remarkably novel phenomena. In order to proceed, the onlyprice to pay is a willingness to give up a large body of conventional tools and intu-ition in favor for building new techniques from the ground up. The task is arduous,but in the end appears fruitful, as our principle dynamical result suggests.
ERAL CURVES AND MINIMAL SETS 11
Overview of Results.
The purpose of this section is to provide an overviewof the most important results proved in this manuscript. The first result, Theorem1, has already been stated, but we restate it here for completeness. The second,Theorem 2, is an immediate generalization. Each of these results are proved inSection 3, however they each rely on some rather non-trivial properties of pseu-doholomorphic curves which are then proved in Section 4. Indeed, these resultsregarding properties of the so-called feral curves appear to be quite fundamentalnot just to our results, but for many future results as well. Indeed, they appear toform the basic foundational analysis for the extension of pseudoholomorphic curvetheory beyond symplectizations of contact and stable Hamiltonian manifolds, andinto the realm of only framed Hamiltonian manifolds and symplectic cobordismswith simply generic boundary. As such, we designate these results as theorems andhighlight them below. In order to understand the statement of some of these results,we also provide some basic definitions, including the namesake of this manuscript,the feral curve . For each such result we provide a brief description to highlight itsutility.
Theorem 1 (main dynamical result) . Consider R equipped with the standard symplectic structure and a Hamiltonian H ∈ C ∞ ( R , R ) for which M := H − (0) is a non-empty compact regular energylevel. Then the Hamiltonian flow on M is not minimal. This of course is the main dynamical result of this manuscript. It is worth notingthat it is crucial that the energy level be compact.
Theorem 2 (second main dynamical result) . Let ( M ± , η ± ) be a pair of compact three-dimensional framed Hamiltonian mani-folds, and let ( (cid:102) W , ˜ ω ) be a symplectic cobordism from ( M + , η + ) to ( M − , η − ) in thesense of Definition 2.11. Suppose that ( (cid:102) W , ˜ ω ) is exact, M − is connected, and that ( M + , η + ) is contact type and has a connected component M (cid:48) which is either S ,overtwisted, or there exists an embedded S in M (cid:48) ⊂ ∂ (cid:102) W which is homotopicallynontrivial in (cid:102) W . Then the flow of the Hamiltonian vector field X η − on M − is notminimal. This is the second main dynamical result of this manuscript. It is perhapssurprising that this generalization can be obtained with so little modifications fromthe proof of Theorem 1.
Remark 1.2 (removing the exactness condition) . It should be straightforward to generalize the argument of the proof to the case whereexactness is replaced by the assumption that (cid:101) ω vanishes on π . This assumptionwould prevent a certain type of bubbling. Possibly, using polyfold technology, onemight even get away without any assumption on the symplectic form (cid:101) ω . We now turn our attention to providing some definitions, which will in turnallow us to state a number of properties of the pseudoholomorphic curves to bestudied. We begin with the notion of a generalized puncture, which is necessary todefine since a priori our curves may be non-compact but their domains need not beconformally equivalent to a finitely punctured Riemann surface.
Definition 1.3 (generalized punctures) . Let S and W each be smooth finite dimensional manifolds, each possibly non-compact, and each possibly with smooth compact boundary. Let u : ( S, ∂S ) → ( W, ∂W ) be a smooth proper map. Let W k ⊂ W be a sequence of open sets eachwith compact closure which satisfy(1) W k ⊂ W k +1 for all k ∈ N (2) W = ∪ k ∈ N W k .Define Punct W k ( S ) to be the number of non-compact path-connected components ofthe set S \ u − ( W k ) . Define Punct( S ) := lim k →∞ Punct W k ( S ) . Remark 1.4 (monotonicity of Punct) . Regarding Definition 1.3, we note that if W (cid:48) ⊂ W (cid:48)(cid:48) are open subsets of W , eachwith compact closure, then it straightforward to show that Punct W (cid:48) ( S ) ≤ Punct W (cid:48)(cid:48) ( S ) and hence Punct( S ) is well-defined, and defined independent of the choice of ex-hausting sequence { W k } k ∈ N . Next we aim to provide the primary novel definition of this manuscript, howeverit relies on a number of standard notions which some readers may not be familiarwith, but which are provided later in Section 2. As such, we note that it will behelpful to be familiar with the notion of a framed Hamiltonian manifold (Definition2.4), an η -adapted almost Hermitian structure (Definition 2.5), and a proper markednodal pseudoholomorphic curve (Definition 2.30). With these understood, we canthen define a feral curve. Definition 1.5 (feral curves) . Let ( M, η ) be a framed Hamiltonian manifold, and let ( J, g ) be an η -adapted almostHermitian structure on R × M . Let u = ( u, S, j, W, J, µ, D ) be a proper markednodal pseudoholomorphic curve (possibly with compact boundary) in R × M . Wesay u is a feral pseudoholomorphic curve , or simply a feral curve , provided(1) (cid:82) S u ∗ ω < ∞ (2) Genus( S ) < ∞ (3) Punct( S ) < ∞ ; that is, ( u, S, j ) has a finite number of generalized punc-tures.(4) µ < ∞ (5) D < ∞ (6) π ( S ) < ∞ The above is the namesake definition of this manuscript. It may be helpful tothink of such a curve simply as being proper pseudoholomorphic map, with finite ω -energy, and finite topology. It is also worth noting that in the more usual casethat η = ( λ, dλ ) is a contact manifold, a feral curve is nothing other than a finiteenergy pseudoholomorphic curve.We are now prepared to state the main properties of feral curves. Theorem 3 (area bounds) . Let ( M, η ) be a compact framed Hamiltonian manifold, let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M , and fix positive constants r > and E > .Then there exists a constant C = C ( J, g, ω, λ, r, E ) with the following property.For each proper pseudoholomorphic map u : S → R × M without boundary which ERAL CURVES AND MINIMAL SETS 13 satisfies (cid:90) S u ∗ ω ≤ E < ∞ , and for which there exists there exists a ∈ R such that ( a ◦ u ) − ( a ) = ∅ (e.g. if a ◦ u ⊂ [0 , ∞ ) × M ), the following holds. (cid:90) (cid:101) S u ∗ ( da ∧ λ + ω ) ≤ C, where (cid:101) S := { ζ ∈ S : a − r < a ◦ u ( ζ ) < a + r } . To be clear: C depends on ambient geometry in R × M , r , and the ω -energy bound E , but not the map u . The above estimate, as well as the generalizations provided in Section 4.1, arerather interesting. Roughly the above states that if a feral curve has a local max-imum or a local minimum, then the area cannot be arbitrarily large in a boundedneighborhood of that extremal point. This is important because in general feralcurves definitely can develop unbounded local area, but in some sense this mustoccur very far away from the absolute minimum or maximum. In Section 4.1 weshall prove a more general result about area bounds in a neighborhood of a levelset of a proper curve with finite ω -energy. Very roughly, we show that for (cid:101) S r := { ζ ∈ S : a − r < a ◦ u ( ζ ) < a + r } . we have Area u ∗ g ( (cid:101) S r ) ≤ Ae Br where A depends on (cid:82) ( a ◦ u ) − (0) u ∗ λ and (cid:82) S u ∗ ω , and B depends only on the geom-etry of the ambient manifold. Theorem 4 ( ω -energy threshold) . Let ( M, η = ( λ, ω )) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adapted almost Hermitian structure on R × M . Also, fix positive constants r > ,and C g > . Then there exists a positive constant < (cid:126) = (cid:126) ( M, η, J, g, r, C g ) withthe following significance. Let { h k } k ∈ N be a sequence of quadruples ( J k , g k , λ k , ω k ) with the property that each η k = ( λ k , ω k ) is a Hamiltonian structure on M , andeach ( J k , g k ) is an η k -adapted almost Hermitian structure on R × M , and supposethat ( J k , g k , λ k , ω k ) → ( J, g, λ, ω ) in C ∞ as k → ∞ . Furthermore, fix a ∈ R , and let u k : S k → R × M be a sequence of compactconnected generally immersed pseudoholomorphic maps which satisfy the followingconditions:( (cid:126)
1) either a ◦ u k ( S k ) ⊂ [ a , ∞ ) or a ◦ u k ( S k ) ⊂ ( −∞ , a ] for all k ∈ N ( (cid:126) Genus( S k ) ≤ C g ( (cid:126) a ◦ u k ( ∂S k ) ∩ [ a − r, a + r ] = ∅ ( (cid:126) a ∈ a ◦ u k ( S k ) .Then for all sufficiently large k ∈ N we have (cid:90) S k u ∗ k ω k ≥ (cid:126) . Whereas Theorem 3 is concerned with showing that the area near an absolutemaximum or minimum of a feral curve cannot be too large, Theorem 4 shows thatit cannot be to small either; or more precisely that the ω -energy cannot be toosmall. This result is one of the easiest to obtain, and follows essentially from acompactness theorem. However, we note that such a compactness theorem requires an area bound which one only has as an application of Theorem 3. We also notethat the bound on genus can almost certainly be removed. Indeed, whereas ourproof employs target-local Gromov compactness, which requires the genus bound,one could probably replace our argument with Taubes’s convergence as integralcurrents which does not require a genus bound. Because some of our later resultsdo require such a genus bound, such a (potentially) superfluous condition is not ahindrance and creates a more self-contained presentation. Theorem 5 (asymptotic connected-local area bound) . Let ( M, η ) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . Then there exists a positive constant r = r ( M, η, J, g ) with the following significance. For each generally immersed feralpseudoholomorphic curve ( u, S, j ) in R × M , there exists a compact set of the form K := [ − a , a ] × M with the property that for each ζ ∈ S such that u ( ζ ) / ∈ K wehave Area u ∗ g (cid:0) S r ( ζ ) (cid:1) ≤ here S r ( ζ ) is defined to be the connected component of u − ( B r ( u ( ζ ))) containing ζ , and B r ( p ) is the open metric ball of radius r centered at the point p ∈ R × M . Arguably, Theorem 5 is the most important property of feral curves developedhere. The difficulty is that in general the Hofer-energy of a feral curve may beinfinite, which is to say that in general, we havesup z ∈ S Area u ∗ g (cid:16) u − (cid:0) B (cid:15) ( u ( z )) (cid:1)(cid:17) = ∞ for each (cid:15) >
0; here B (cid:15) ( p ) is a ball of radius (cid:15) centered at p ∈ R × M . In contrast,Theorem 5 states that if we replace u − ( B (cid:15) ( u ( z ))) with the connected componentin this set containing z , then the associated supremum is in fact finite. Establishingthis estimate is a rather technical process, and it is perhaps worth noting that ourproof crucially relies on the fact that the genus of a feral curve is finite and that thenumber of generalized punctures is also finite. Indeed, deducing these area boundsin part from genus bounds is notably delicate.With such a “connected-local” area bound established, a variety of asymptoticproperties of feral curves can be established essentially via target-local Gromovcompactness. One such important result is the following. Theorem 6 (asymptotic curvature bound) . Let ( M, η ) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . For each feral pseudoholomorphic curve u =( u, S, j, R × M, J, µ, D ) , there exists a compact set of the form K := [ − a , a ] × M ,and positive constant C κ = C κ ( M, η, J, g ) with the following significance. First, therestricted map u : S \ u − ( K ) → R × M is an immersion. Second, for each ζ ∈ S \ u − ( K ) we have (cid:107) B u ( ζ ) (cid:107) ≤ C κ ERAL CURVES AND MINIMAL SETS 15 where B u ( ζ ) is the second fundamental form of the immersion u evaluated at thepoint ζ . It may be useful to paraphrase the above result as saying that outside a largecompact set, a feral curve is immersed with a uniform point-wise curvature bound.This result, together with Theorem 5 guarantees that our curves have the nicestpossible asymptotic behavior given that the Hofer-energy can be infinite. Indeed,given that a curve with infinite Hofer-energy is generally thought to be too wild toanalyze, the above two results provide a tremendous amount of structure.
Theorem 7 (existence workhorse) . Let ( M, η ) be a compact framed Hamiltonian manifold with dim( M ) = 3 . Let { a k } k ∈ N ⊂ R − be a sequence for which a k → −∞ monotonically. For each k ∈ N ,let ( J k , g k ) be a η -adapted almost complex structure on R × M . Suppose that thereexists a positive constant C ≥ , and suppose that for each k ∈ N and each b ∈ [ a k , there exists a stable unmarked but possibly nodal pseudoholomorphic curve u bk = (cid:0) u bk , S bk , j bk , ( −∞ , × M, J k , ∅ , D bk (cid:1) with the following properties. (P1) the topological space | S bk | is connected, which implies (P4) below,(P2) u bk is compact and u bk ( ∂S bk ) ⊂ (0 , × M ,(P3) inf ζ ∈ S bk a ◦ u bk ( ζ ) = b ,(P4) there exists a continuous path α : [0 , → | S bk | satisfying a ◦ u bk ◦ α (0) = b and α (1) ∈ ∂S bk ,(P5) Genus( S bk ) ≤ C ,(P6) (cid:82) S bk ( u bk ) ∗ ω ≤ C ,(P7) D bk ≤ C ,(P8) the number of connected components of ∂S bk is bounded above by C .Furthermore, suppose that J k → J in C ∞ , and for each fixed k , and each pair b, b (cid:48) ∈ [ a k , with b (cid:54) = b (cid:48) we have (cid:0) u bk ( S bk ) ∩ u b (cid:48) k ( S b (cid:48) k ) (cid:1) ≤ C. Then there exists a closed set Ξ ⊂ M satisfying ∅ (cid:54) = Ξ (cid:54) = M which is invariantunder the flow of the Hamiltonian vector field X η . Our final main result regarding feral curves is the above workhorse theorem,which perhaps requires some explanation. First, we must note how much morecomplicated this result is than the finite energy case. Indeed, in the contact caseit is sufficient to know that a finite energy curve exists in order to deduce that aperiodic orbit exists. In contrast, we cannot guarantee a similar dynamics resultin the case of having a feral curve. The trouble is that although a feral curve doesindeed have a notion of a limit set, which is both closed and invariant, in general itmay be the entire framed Hamiltonian manifold M . That is, the main difficulty isto establish that the limit (or rather, a limit) of a feral curve is not all of M , andthis is why we need Theorem 7. While of course it would be more appealing to have Here we mean stable in the sense described just after Definition 2.30. By definition, the u bk are boundary-immersed. conditions on a single feral curve which guaranteed that the associated limit set wasnot the total space, Theorem 7 is general enough to apply to a rather large numberof cases in which one constructs feral curves, and hence establishes non-minimaldynamics in a good number of cases.As we close out this section, we provide a brief synthesis of the above results, inorder to illustrate the new class of curves that have been found. Specifically, we putferal curves in the context of some historical curves as well as a more general classwhich we introduce as F -dominated curves. As we shall see, these F -dominatedcurves have a weak but useful notion of a compactness result. To define them, itwill be convenient to make the following preliminary definition for a proper map u : S → R × M . Indeed, for each c ∈ R and r >
0, we define S r ( c ) := u − (cid:0) I r ( c ) × M (cid:1) where I r ( c ) := [ c − r, c + r ] ⊂ R . Definition 1.6 ( F -dominated pseudoholomorphic curves) . Let ( M, η ) be an almost Hermitian manifold equipped with an η -adapted almostHermitian structure. Let u = ( u, S, j, R × M, J, µ, D ) be a marked nodal pseudo-holomorphic curve . Suppose further that u : S → R × M is proper, and for eachconnected component S (cid:48) ⊂ S on which the restriction u : S (cid:48) → R × M is constant,we have χ ( S (cid:48) ) − (cid:0) S (cid:48) ∩ ( µ ∪ D )) < . Let F : R → [0 , ∞ ) be a continuous function. We say u is F -dominated, providedthere exists an c ∈ R such that the following holds for every r ∈ R : Points (cid:0) S r ( c ) (cid:1) + Genus (cid:0) S r ( c ) (cid:1) + Area u ∗ g (cid:0) S r ( c ) (cid:1) ≤ F ( r ) where Points (cid:0) S r ( c ) (cid:1) := (cid:0) ( µ ∪ D ) ∩ S r ( c ) (cid:1) . Geometrically, we note that any slightly reasonable proper pseudoholomorphicmap u : S → R × M is F -dominated for some F . In contrast, if first given an F ,one next finds a curve which is in fact F -dominated, then this condition guaranteesa certain maximal growth rate of the area as one moves away from a reference level(say { } × M ). Similarly, we have bounds on the growth rate of genus and specialpoints etc. Next, let us recall that when Gromov introduced pseudoholomorphiccurves in closed symplectic manifolds, a taming condition guaranteed that curvesin a fixed homology class had uniformly bounded total area. Moreover, a uniform total area bound was precisely the analytic condition needed to prove compactnessof a family of curves. Then, in the symplectization case for SFT, specifically withan R -invariant Riemannian metric, it turned out that curves in a fixed (relative)homology class could develop infinite area, however they still had a uniform local area bound. Again, this uniform local area bound was precisely the conditionneeded to prove SFT compactness. Feral curves then go one step further in thisprogression, and need not even have uniform local area bounds. Instead they haveuniform connected local area bounds, as in Theorem 5, and moreover fit withinthe class of F -dominated pseudoholomorphic curves, and hence one can prove asort weak one-level SFT compactness theorem, or more specifically an exhaustive Gromov compactness theorem; see Definition 2.38 and Theorem 2.39 below. See Definition 2.3. See Definition 2.5. See Definition 2.30.
ERAL CURVES AND MINIMAL SETS 17
We can then summarize as follows. The results in our paper show that feralcurves belong to the distinguished class of F -dominated pseudoholomorphic curvesfor which a version of an exhaustive Gromov compactness theorem exists, see [13].However, feral curves are somewhat more special:(1) Due to the assumption of finiteness of the ω -energy and a bound on genusand the number of ends, it follows that the behavior of the ends of thesecurves reflects some of the underlying dynamical features of the Hamilton-ian flow on M .(2) In general, the uniform connected-local area bound can be obtained fromtopological bounds, which is an important feature in their construction.We note that the methods in this paper can be used to establish the existence ofnontrivial feral curves in quite general contexts. We now take a moment to describethis process in a fairly general setting, thereby establishing truly feral curves. RecallProposition 2.8 which associates to a smooth compact regular energy surface M ofa Hamiltonian function H on a symplectic manifold ( W, Ω) the data ( λ, ω, J ) whichequips R × M with a canonical almost complex structure. For simplicity assumethat M is connected and denote the closures of the two components of W \ M by by A and B so that A ∩ B = M . In favorable circumstances, for example a sufficientlyrich Gromov-Witten theory, one can use a stretching construction around M asdescribed in this paper to obtain feral curves in R × M with image in [0 , ∞ ) × M or ( −∞ , × M . Specifically carrying out this idea in the following example leadsto true feral curves which are not of finite energy. Pick a smooth compact regularhypersurface M in the standard symplectic vector space R n , with n ≥
3, whichdoes not admit a periodic orbit. Such surfaces exist by the results of M. Hermanand Ginzburg, see [15, 16, 22]. One can view M as lying in CP n in the complementof the divisor at infinity. Then the fact that there are plenty of complex lines al-lows one to carry out the previously described deformation argument in such a waythat the obtained curve is feral, and even better has a R -projection which has aminimum.As a final remark, we would like to forewarn the reader that we have meticu-lously kept track of constants in our estimates, and that for the first ninety pagesit would seem that the sole purpose of this is to torture the reader. However, inlater proofs we use rather sophisticated arguments which only work because of ourcareful book keeping. Acknowledgements:
The first author would like to thank Professors Kai Cieliebak,William Minicozzi, and Chris Wendl for a number of helpful conversations. The firstauthor would also like to thank the Institute for Advanced Study, the Universityof Massachusetts Boston, and the National Science Foundation for their generousaccommodation and support of this research.2.
Background
In this section, we will recall some basic notions which will be used through-out later sections. All of these definitions should be either well known or readilyabsorbed by specialists of pseudoholomorphic curves. We note that there are twonotions presented below which may nevertheless be unfamiliar to such a reader,namely namely so-called target-local Gromov compactness (see Theorem 2.36) and exhaustive Gromov convergence (see Definition 2.38) and compactness (see The-orem 2.39). Before progressing to those rather technical concepts, we begin withmore elementary notions.
Remark 2.1 (on smoothness) . Throughout this article, when referring to the regularity of differentiable objects(functions, forms, manifolds, etc.) the term smooth will always refer to C ∞ -smooth;any less regularity, for example C , will be mentioned explicitly. Ambient Geometric Structures.
Here we begin by considering geometricstructures on certain manifolds which will serve as the target space for our laterdefined pseudoholomorphic maps.
Definition 2.2 (almost complex manifold) . Let W be a smooth manifold not necessarily closed, possibly with boundary, and let J ∈ Γ(End(TW)) be a smooth section for which J ◦ J = − . We call J an almostcomplex structure for W , and the pair ( W, J ) an almost complex manifold. Definition 2.3 (almost Hermitian manifold) . Let W be a smooth finite dimensional manifold equipped with an almost com-plex structure J and a Riemannian metric g . We say the pair ( J, g ) is an al-most Hermitian structure on W provided that J is an isometry for g . That is, g ( x, y ) = g ( Jx, Jy ) for all x, y ∈ T W . We pause for a moment to comment on almost Hermitian manifolds, since theymay at first seem needlessly tangential to the more natural objects of study, namelysymplectic manifolds with compatible or tame almost complex structures. First wepoint out that any almost complex manifold (
W, J ) can be given an almost Her-mitian structure (
J, g ) by choosing an arbitrary Riemannian metric ˜ g and defining g ( x, y ) := (cid:0) ˜ g ( x, y ) + ˜ g ( Jx, Jy ) (cid:1) . Second, for a symplectic manifold ( W, Ω), an Ω-compatible almost complex structure J satisfies, by definition, the property that g ( x, y ) := Ω( x, Jy ) is a Riemannian metric. For this metric, we immediately seethat J is a g -isometry. Third, in the case that J is Ω-tame, we have Ω( x, Jx ) > x ∈ T W with x (cid:54) = 0. An associated Riemannian metric is then given by g ( x, y ) = (cid:0) Ω( x, Jy ) + Ω( y, Jx ) (cid:1) , for which again J a g -isometry.At this point, one may still question the utility of moving from analysis in sym-plectic manifolds to almost Hermitian manifolds, and the answer is fairly simple:The manifolds that occupy our primary interest are not, strictly speaking, symplec-tic. Moreover, the principle role a symplectic form typically plays is to guaranteethat pseudoholomorphic curves (defined below) in a fixed homology class have uni-formly bounded energy, or area; however, this a priori bound fails in the almostHermitian manifolds in which we are interested. Nevertheless, our analysis willrequire the aid of Riemannian metric for which the almost complex structure is anisometry. This inevitably leads the definition of an almost Hermitian manifold, andhence motivates our generalization.In order to more precisely specify the manifolds of interest, we will require twomore definitions. Definition 2.4 (framed Hamiltonian structure) . Let M be a n + 1 dimensional closed manifold, and let λ and ω respectively be asmooth one-form and smooth two-form on M . We say η := ( λ, ω ) is a Hamiltonianstructure for M provided dω = 0 and λ ∧ ω n is a volume form on M . We call ( M, η ) ERAL CURVES AND MINIMAL SETS 19 a framed Hamiltonian manifold . We call ( M, η ) an exact framed Hamiltonianmanifold, and η = ( λ, ω ) an exact Hamiltonian structure, provided there is a one-form τ on M , for which ω = dτ . Note that in the special case that a framed Hamiltonian structure ( λ, ω ) satisfiesthe additional condition that ker ω ⊂ ker dλ , we call ( λ, ω ) a stable Hamiltonianstructure. Throughout this article we will not make this additional assumption,however it will often be useful to make comparisons between analysis in the framedversus stable case. We also note that there exists a vector field X η associated to aHamiltonian structure η = ( λ, ω ) uniquely determined by the equations λ ( X η ) ≡ ω ( X η , · ) ≡ . We call X η the Hamiltonian vector field associated to η . Observe that by definitionof X η and Cartan’s formula, we have: L X η ω = d ( i X η ω ) + i X η dω = 0 . Definition 2.5 ( η -adapted almost Hermitian structures) . Given a manifold M with a framed Hamiltonian structure η = ( λ, ω ) , we consider R × M ; we henceforth equip R with the coordinate a . Furthermore, we say the pair ( J, g ) is an η -adapted almost Hermitian structure on R × M provided it satisfies thefollowing conditions.(J1) J is an R -invariant almost complex structure(J2) J∂ a = X η (J3) g = ( da ∧ λ + ω )( · , J · ) is a Riemannian metric.(J4) J : ker λ ∩ ker da → ker λ ∩ ker da where we have abused notation by writing λ and ω instead of pr ∗ λ and pr ∗ ω where pr : R × M → M is the canonical projection. We shall refer to R × M colloquiallyas the ‘symplectization’ of M even if this is admittedly not a good name. We will need to verify that our definition of η -adapted almost Hermitian structureis aptly named. Specifically, we will need to verify that J is indeed an isometry for g . This will be accomplished momentarily, see Lemma 2.7 below, however first weneed the following. Lemma 2.6 (property of η -adapted J ) . Let J be an adapted almost complex structure on the symplectization of the manifold M with Hamiltonian structure ( λ, ω ) . Then − da ◦ J = λ and ω ( Y, JY ) ≥ for all Y ∈ T ( R × M ) . Proof.
Observe that any tangent vector Y ∈ T ( R × M ) can be uniquely written as Y = c ∂ a + c X η + Y ξ with Y ξ ∈ ker da ∩ ker λ . In this case we have − da ◦ J ( Y ) = − da ( c J∂ a + c JX η + JY ξ ) = − da ( c X η − c ∂ a + JY ξ ) = c = λ ( Y ) . To prove the second part, we compute as follows. ω (cid:0) c ∂ a + c X η + Y ξ , J ( c ∂ a + c X η + Y ξ ) (cid:1) = ω ( Y ξ , JY ξ )= ( da ∧ λ + ω )( Y ξ , JY ξ )= (cid:107) Y ξ (cid:107) g ≥ . (cid:3) Lemma 2.7 ( η -adapted ( J, g ) are indeed almost Hermitian) . Let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) , and let ( J, g ) be an η -adapted almost Hermitian structure on R × M in the sense of Definition 2.5. Then (
J, g ) is an almost Hermitian structure for R × M . That is, J is an isometryfor g .Proof. By Definition 2.5, we know that g = ( da ∧ λ + ω )( · , J · ) is a Riemannianmetric, so we must show that g ( X, Y ) = g ( JX, JY ). Observe that g ( X, Y ) = ( da ∧ λ + ω )( X, JY )= da ( X ) λ ( JY ) − λ ( X ) da ( JY ) + ω ( X, JY ) , and by Lemma 2.6, we have − da ( J · ) = λ ( · ) so λ ( J · ) = da ( · ), from which it imme-diately follows that(1) g = da ⊗ da + λ ⊗ λ + ω ( · , J · )and g ( JX, JY ) = da ( JX ) λ ( JJY ) − λ ( JX ) da ( JJY ) + ω ( JX, JJY ) , = − da ( JX ) λ ( Y ) + λ ( JX ) da ( Y ) − ω ( JX, Y ) , = da ( Y ) λ ( JX ) − λ ( Y ) da ( JX ) + ω ( Y, JX ) , = g ( Y, X )= g ( X, Y )so indeed, J is an isometry, and hence ( J, g ) is an almost Hermitian structure. (cid:3)
We take a moment to give an example of how a framed Hamiltonian structureand adapted almost complex structure might arise in practice. Indeed, we firstconsider a (2 n +2)-dimensional symplectic manifold ( W, Ω) equipped with a smoothfunction H : W → R for which 0 is a regular value. One may allow that W is amanifold with boundary, however in this case we require that { H = 0 } ∩ ∂W = ∅ Assume further that J is an almost complex structure on W for which Ω( · , J · )is a Riemannian metric. Define M := H − (0), and consider ξ := T M ∩ JT M as a subset of
T M ⊂ T W . A bit of linear algebra shows that ξ is a hyperplanedistribution in T M , and by construction J : ξ → ξ . Next, define the vector field X H ∈ Γ( T M ) by the following Ω( X H , · ) = − dH. Note that X H never vanishes since 0 is a regular value of H . We then define λ on M to be the unique one-form for which λ ( X H ) ≡ λ = ξ. Regarding M as a closed manifold with i : M (cid:44) → W the canonical inclusion, wedefine ω to be the closed two-form given by ω := i ∗ Ω. To show that ( λ, ω ) isa framed Hamiltonian structure for M , it is then sufficient to show that λ ∧ ω n is a volume form on M . Since J : ξ → ξ , and Ω( · , J · ) is a Riemannian metric,and since Ω (cid:12)(cid:12) ξ = ω , it follows that there exists a symplectic basis of ξ of the form { e , Je , . . . , e n , Je n } . However, we then have λ ∧ ω n ( X H , e , Je , . . . , e n , Je n ) = λ ( X H ) · ω n ( e , Je , . . . , e n , Je n ) > ERAL CURVES AND MINIMAL SETS 21 here we have made use of the fact that λ (cid:12)(cid:12) ξ ≡
0, and for any v ∈ ξ we have ω ( X H , v ) = Ω( X H , v ) = − dH ( v ) = 0 since ξ ⊂ T M and dH (cid:12)(cid:12) T M ≡
0. Thus ( λ, ω ) isindeed a framed Hamiltonian structure for M . Using the fact that 0 = − dH (cid:12)(cid:12) T M =Ω( X H , · ) (cid:12)(cid:12) T M = ω ( X H , · ) and the definition of λ , we find that X H = X η .Continuing our construction, we obtain an adapted almost complex structureon R × M in the following way. We demand R -translation invariance, so it issufficient to define J along { } × M ⊂ R × M . Along this hypersurface, we canidentify ξ ⊂ T M with ker da ∩ ker λ ⊂ T ( { } × M ). Using this identification, J : ker da ∩ ker λ → ker da ∩ ker λ , and we then define J∂ a = X H = X η .We summarize the previous discussion as follows. Proposition 2.8 (energy levels are framed Hamiltonian) . Consider a symplectic manifold ( W, Ω) equipped with a compatible almost complexstructure J , and a smooth function H : W → R for which is a regular valueand M := H − (0) is compact and disjoint from ∂W . Then M naturally carriesa framed Hamiltonian structure η = ( λ, ω ) defined by ker( λ ) = T M ∩ J ( T M ) and λ ( X H ) ≡ , where ω is the pull-back of Ω to M . With the restriction of J to T M ∩ J ( T M ) denoted again by J , we obtain ( λ, ω, J ) which defines a natural η -adapted almost Hermitian structure ( J, g ) on the symplectization R × M . We now provide several more general target manifolds with adapted structures,the first of which we call a realized Hamiltonian homotopy , and which is made precisein Definition 2.9 below. For clarity, we first provide the following motivation forthe definition. Consider a closed symplectic manifold ( W, Ω) with compatible J asin Proposition 2.8, and consider a smooth Hamiltonian H : W → R for which 0 is aregular value. Let us define M := H − (0), and observe that M is diffeomorphic to { H = t } for all t sufficiently close to 0, and an explicit diffeomorphism is obtainedby the flow of the vector field Y := ∇ H (cid:107)∇ H (cid:107) . We denote this diffeomorphism by ψ t : M → { H = t } . As a consequence of Proposition 2.8, this gives rise to a familyof framed Hamiltonian structures η t := ( λ t , ω t ) on M with ω t = ψ ∗ t Ω. Moreover,for all t and t sufficiently close to 0 we in fact have that ( λ t , ω t ) is a framedHamiltonian structure on M . With this in mind, we then consider I ⊂ R to beany open interval, and we let f : I → R be any smooth function mapping into aneighborhood of 0 for which f (cid:48) ≥
0. We can then equip
I × M with the one-formˆ λ := π ∗ λ t and with the two-form ˆ ω = Ψ ∗ Ω where Ψ( t, p ) = ψ f ( t ) ( p ). In this way,a homotopy of two-forms t (cid:55)→ ω t on M arising from framed Hamiltonian structuresgives rise to a single two-form ˆ ω on I × M , and somewhat similarly for ˆ λ . It is forthis reason that we call ( I × M, (ˆ λ, ˆ ω )) a “realized Hamiltonian homotopy.” Wemake this idea both more precise and more general with the following definition. Definition 2.9 (realized Hamiltonian homotopy) . Let M be a smooth (odd-dimensional) closed manifold, let I ⊂ R be an intervalequipped with the coordinate t , and let ˆ λ and ˆ ω respectively be a one-form and two-form on I × M . We say ( I × M, (ˆ λ, ˆ ω )) is a realized Hamiltonian homotopy providedthe following hold.(1) ˆ λ ( ∂ t ) = 0 .(2) i ∂ t ˆ ω = 0 .(3) d ˆ ω (cid:12)(cid:12) { t =const } = 0 (4) dt ∧ ˆ λ ∧ ˆ ω ∧ · · · ∧ ˆ ω > . (5) ˆ λ is invariant under the flow of ∂ t (6) if I is unbounded, then there exists a neighborhood of {±∞} × M on which ˆ ω is invariant under the flow of ∂ t . We note that the properties of a realized Hamiltonian homotopy imply that,near {±∞} × M , ˆ λ and ˆ ω are pull-backs of forms λ and ω on M , where in addition ω is closed and further λ ∧ ω n > n + 1 = dim( M ). Observe that arealized Hamiltonian homotopy gives rise to two additional structures, the first ofwhich is a vector field (cid:98) X on I × M which is uniquely determined by the equations dt ( (cid:98) X ) = 0 , ˆ λ ( (cid:98) X ) = 1 , i (cid:98) X ˆ ω = 0 . The second structure is the codimension-two plane field distribution given byˆ ξ = ker dt ∩ ker ˆ λ. With this in mind, we now provide the notion of an almost Hermitian structureadapted to a realized Hamiltonian homotopy.
Definition 2.10 (adapted structures for a realized Hamiltonian homotopy) . Let ( I × M, (ˆ λ, ˆ ω )) be a realized Hamiltonian homotopy. We say an almost Her-mitian structure ( (cid:98) J, ˆ g ) on I × M is adapted to this realized Hamiltonian homotopyprovided the following hold.(1) (cid:98) J∂ t = (cid:98) X .(2) (cid:98) J : ˆ ξ → ˆ ξ .(3) ˆ g = ( dt ∧ ˆ λ + ˆ ω )( · , (cid:98) J · ) .(4) if I is unbounded, then there exists a neighborhood of {±∞} × M on whichthe restriction (cid:98) J (cid:12)(cid:12) ˆ ξ is invariant under the flow of ∂ t . Definition 2.11 (symplectic cobordism) . Let M + and M − be smooth closed manifolds, and let η ± = ( λ ± , ω ± ) denote framedHamiltonian structures for each. Suppose that each M ± is oriented by the volumeform λ ± ∧ ω ± ∧ · · · ∧ ω ± . We say a compact symplectic manifold ( (cid:102) W , ˜ ω ) is a symplectic cobordism from ( M + , η + ) to ( M − , η − ) provided(1) ∂ (cid:102) W = M + − M − (2) i ∗± ˜ ω = ω ± , where i ± : M ± → (cid:102) W is the canonical inclusion. The basic concept we would like to introduce next is that of an extended sym-plectic cobordism and it is quite involved, however the overall notion should bemostly familiar to those experienced with symplectic manifolds with cylindricalends. The complexity of the definition arises in large part because the ends arenot (symplectizations of) contact or stable Hamiltonian manifolds, and hence laterwhen we derive new energy/area estimates for pseudoholomorphic curves we mustrely on a very carefully arranged structure in the target manifold, specifically onthe region which transitions from symplectic part to the cylindrically ended part.To help digest these notions, we break the definition apart and introduce auxil-iary objects in several preliminary definitions. In the first definition we disregard
ERAL CURVES AND MINIMAL SETS 23 _1-1 ε−1+ε1−(1/4)ε1−(1/8)ε 0−1+(1/8)ε−1+(1/4)ε Cyl+Cyl-Core
Figure 1.
The figure shows the core, Cyl + and Cyl − regions andindicates the region where d ¯ a ( ∂ ¯ a ) = 1.symplectic and almost complex considerations and target the notion of an extended cobordism in general. Definition 2.12 (extended cobordism) . Let M ± be two closed oriented manifolds. An extended cobordism from M + to M − is given by a tuple ( ¯ W, ¯ a, ∂ ¯ a , ε ) and embeddings φ ± : M ± → ¯ W with the followingproperties.(1) W is a smooth oriented manifold.(2) ¯ a : W → R is a proper smooth function.(3) ∂ ¯ a ∈ Γ( T W ) is a smooth complete vector field.(4) ε > .This data has the following additional properties:(1) d ¯ a ( ∂ ¯ a ) = 1 on the domain {| ¯ a | > − ε } .(2) The images of the embeddings φ ± are the codimension-one submanifolds ¯ a − ( ± .(3) The orientation induced on ¯ a − (1) ∪ ¯ a − ( − as the boundary of the sub-domain with smooth boundary ¯ a − ([ − , of W has the property that φ + : M + → ¯ a − (1) is orientation-preserving and φ − : M − → ¯ a − ( − is orientation-reversing.The compact sub-domain with smooth boundary defined by Core( W ) := (cid:8) | ¯ a | ≤ − ε (cid:9) (2) is called the “core of W .” The following domains are called the “positive cylinder”and “negative cylinder” respectively. Cyl ± ( W ) := (cid:8) ± ¯ a > − ε (cid:9) . (3)We illustrate the above definition by Figure 2.1. We also note that in the casethat ε > d ¯ a ( ∂ ¯ a ) = 1 on all of W , and hence in this case W is diffeomorphic to the oriented product R × M + via ( s, m ) → s · φ + ( m ), where( s, w ) → s · w is notation for the flow associated to ∂ ¯ a .Definition 2.12 above has introduced the “smooth aspects” of the target mani-folds which we shall need for further analysis, and so our next step is to introducethe additional symplectic, Hermitian, and framed Hamiltonian features on suchmanifolds. In this case we shall start with ( M ± , η ± ), where η ± = ( λ ± , ω ± ) areframed Hamiltonian structures, and we will define an extended symplectic cobor-dism between ( M + , η + ) and ( M − , η − ) which carries additional compatible data.For this new extension, with underlying extended cobordism ( W , ¯ a, ∂ ¯ a , ε ), it will beimportant to identify certain subdomains distinguished by ranges of values of thefunction ¯ a . While the explicit definition of these subdomains will seem pedantic, itis important to realize that they are necessary for the delicate analysis performedlater. In order to aid comprehension of these technical elements, we will elaborateon some of the more important features after providing the definition.As a final point, we mention that our notion of an extended symplectic cobor-dism actually relies on not just a symplectic structure, but also framed Hamiltonianstructures and an adapted almost Hermitian structure. As such, it would be moreaccurate to give it a more cumbersome name like “extended almost K¨ahler cobor-dism,” however here and throughout we opt for that brevity which emphasizes themost important structure. Definition 2.13 (extended symplectic cobordism) . Consider a pair of framed Hamiltonian structures ( M ± , η ± ) , with η ± = ( λ ± , ω ± ) and view M ± equipped with the induced orientations. Consider also the tuple W =( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) and embeddings φ ± : M ± → W consisting of the following data.(1) ( W , ¯ a, ∂ ¯ a , ε ) together with the φ ± is an extended cobordism between the ori-ented manifolds M + and M − .(2) ¯ ω is a smooth closed two-form on W .(3) ( J, ¯ g ) is an almost Hermitian structure on W .This data has the following additional properties. There exists a smooth function β : R → R satisfying β (¯ a ) = 0 on {| ¯ a | ≥ } β (cid:48) (¯ a ) > on { − (cid:15) < | ¯ a | < } such that the following hold:(1) on the region {| ¯ a | < } we have(a) ¯ ω is non-degenerate.(b) J is ¯ ω -compatible; that is, ¯ ω ( · , J · ) is a Riemannian metric.(2) on the region {| ¯ a | < − (cid:15) } we have ¯ g = ¯ ω ( · , J · ) (3) on the region {| ¯ a | > − (cid:15) } , where we recall that d ¯ a ( ∂ ¯ a ) = 1 , we have(a) ¯ ω = ω ± + d ( βλ ± ) = (cid:16) β (cid:48) ( d ¯ a ∧ λ ± ) (cid:17) + (cid:16) ω ± + βdλ ± (cid:17) where β = β (¯ a ) and we have abused notation by writing λ ± and ω ± rather than themore accurate ( φ − ± ◦ pr ± ) ∗ λ ± and ( φ − ± ◦ pr ± ) ∗ ω ± where pr ± : {± ¯ a > − (cid:15) } → { ¯ a = ± } is the smooth projection along the trajectories of ∂ ¯ a .(b) Make the following definitions:(i) the two form ˆ ω ± := ω ± + βdλ ± ERAL CURVES AND MINIMAL SETS 25 (ii) the co-dimension two plane field ξ := ker d ¯ a ∩ ker λ ± (iii) smooth vector field X determined by d ¯ a ( X ) = 0 λ ± ( X ) = 1 ker ˆ ω ± = Span( ∂ ¯ a , X ); then T W = ξ ⊕ ker ˆ ω ± , and J preserves this splitting; moreover wehave J∂ ¯ a = X , and ˆ ω ± ( · , J · ) (cid:12)(cid:12) ξ is a bundle metric(c) there exists a smooth positive function θ : R → R satisfying θ (¯ a ) = 1 for | ¯ a | > − (cid:15) for which ¯ g = (cid:0) θ (¯ a )( d ¯ a ∧ λ ± ) + ˆ ω ± (cid:1) ( · , J · ) (4) on the region {| ¯ a | ≥ } we have J is invariant under the flow of ∂ ¯ a Then we call the tuple ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) an extended symplectic cobordism from ( M + , η + ) to ( M − , η − ) equipped with adapted almost Hermitian structure. We need a quick definition before we provide some comments on the previousdefinition.
Definition 2.14 (compact region) . Let W be a manifold. Suppose U ⊂ W is an open set for which its closure cl( U ) inherits from W the structure of a smooth compact manifold possibly with boundary.Then we call cl( U ) a compact region in W . Remark 2.15 (structural observations) . In light of the careful definition of an extended symplectic cobordism denoted by W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) , it is possible to identify a number of structures which wewill make use of later. The first of these is the set Core( W ) , as defined in equation(2), and which also is a compact region in the sense of Definition 2.14. Likewise,we also make use of the open regions Cyl ± ( W ) , as defined in equation (3). We alsospecifically note that W = Cyl − ( W ) ∪ Core( W ) ∪ Cyl + ( W ) . Second, we note there exist diffeomorphisms Φ ± : I ± × M ± → Cyl ± ( W )Φ ± ( t, p ) = ϕ t ∓ ∂ ¯ a (cid:0) φ ± ( p ) (cid:1) where I + = (cid:0) − (cid:15), ∞ (cid:1) and I − = (cid:0) − ∞ , − (cid:15) (cid:1) , the φ ± : M ± → { ¯ a = ± } ⊂ W are as in Definition 2.13, and ϕ t∂ ¯ a is the time t flow of the vector field ∂ ¯ a .As a consequence of these structures, we see that the manifolds I ± × M ± equippedwith the pair of differential forms (Φ ∗± λ ± , Φ ∗± ˆ ω ± ) are each a realized Hamiltonianhomotopy in the sense of Definition 2.9; here recall that ˆ ω ± = ω ± + βdλ ± and ¯ ω = ω ± + d ( βλ ± ) for β : R → R satisfying β (¯ a ) = 0 on {| ¯ a | ≥ − (cid:15) } and β (cid:48) (¯ a ) > on the region { − (cid:15) < | ¯ a | < } . Moreover, the pair (Φ ∗± ¯ g, Φ ∗± J ) is an adapted structure for therealized Hamiltonian homotopy, in the sense of Definition 2.10. Finally, regarding
Core( W ) , we note that there exists a positive constant, denoted C θ = C θ ( W ) , for which C − θ ≤ inf ¯ q ∈ Core( W ) inf v ∈ T ¯ q W (cid:107) v (cid:107) ¯ g =1 ¯ ω ( v, Jv ) ≤ sup ¯ q ∈ Core( W ) sup v ∈ T ¯ q W (cid:107) v (cid:107) ¯ g =1 ¯ ω ( v, Jv ) ≤ C θ . (4)The next lemma will give a means to construct an extended symplectic cobordism W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) from a symplectic cobordism (cid:102) W from M + to M − providedthe M ± are equipped with framed Hamiltonian structures. The proof of the lemmawill also make the content and utility of Definition 2.13 more transparent. Lemma 2.16 (cobordism to extended cobordism) . Let ( M ± , η ± ) be a pair of framed Hamiltonian manifolds with η ± = ( λ ± , ω ± ) . Let ( (cid:102) W , ˜ ω ) be a symplectic cobordism from ( M + , η + ) to ( M − , η − ) . Then there existsan extended symplectic cobordism from ( M + , η + ) to ( M − , η − ) equipped with anadapted almost Hermitian structure, which we denote W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) .Moreover, there exists a smooth surjection Ψ : W → (cid:102) W such that the followinghold.(1) the domain restricted map Ψ : {| ¯ a | < } → (cid:102) W \ ∂ (cid:102) W is a diffeomorphism(2) Ψ( {| ¯ a | ≥ } ) = ∂ (cid:102) W (3) ¯ ω = Ψ ∗ (cid:101) ω Proof.
We begin by noting that a version of Darboux’s theorem guarantees thatthere exists an (cid:15) >
0, disjoint neighborhoods (cid:101) O ± of the M ± ⊂ ∂ (cid:102) W in (cid:102) W , anddiffeomorphisms ψ + : ( − (cid:15), × M + → O + and ψ − : [0 , (cid:15) ) × M − → O − for which ψ ∗± ˜ ω = ω ± + d ( tλ ± ) where t is the coordinate on ( − (cid:15),
0] and [0 , (cid:15) ). Thisis the desired positive number (cid:15) > I + := (1 − (cid:15), ∞ ) I − := ( −∞ , (cid:15) − I + := (1 − (cid:15), ∞ ) ˇ I − := ( −∞ , (cid:15) − W as follows. W = (cid:16)(cid:0) I − × M − (cid:1) (cid:116) (cid:102) W (cid:116) (cid:0) I + × M + (cid:1)(cid:17) / ∼ (5)where ˜ q ∼ ( t, p ) provided one of the following two holds:(1) ˜ q ∈ (cid:102) W and ( t, p ) ∈ (1 − (cid:15), × M + and ψ + ( t − , p ) = ˜ q (2) ˜ q ∈ (cid:102) W and ( t, p ) ∈ [ − , (cid:15) − × M − and ψ − ( t + 1 , p ) = ˜ q This defines the desired manifold W . In light of the definition of W , we then let I , I − , and I + denote the natural embeddings: I ± : I ± × M ± (cid:44) → W and I : (cid:102) W (cid:44) → W .
In what follows, it will be convenient to have the following sets established. Foreach δ ∈ [0 , (cid:15) ] we define the positive and negative end regions W + δ = I + (cid:0) (1 − δ, ∞ ) × M + (cid:1) and W − δ = I − (cid:0) ( −∞ , δ − × M − (cid:1) ERAL CURVES AND MINIMAL SETS 27
With these established, we now define a smooth function ¯ a : W → R satisfying¯ a (¯ q ) = (cid:40) pr ◦ I − (¯ q ) if ¯ q ∈ W + (cid:15) pr ◦ I − − (¯ q ) if ¯ q ∈ W − (cid:15) where pr is the canonical projection to the first factor, and | ¯ a (¯ q ) | ≤ − (cid:15) for all ¯ q ∈ W \ (cid:0) W − (cid:15) ∪ W + (cid:15) (cid:1) . This defines the desired function ¯ a . Letting t denote the coordinate on I ± asappropriate, we define ∂ a := ( I ± ) ∗ ∂ t , and smoothly extend it on the rest of W .This defines the smooth vector field ∂ ¯ a , and also establishes property (3a). It alsoallows us to define the desired diffeomorphisms: φ ± : M ± → { ¯ a = ± } φ ± ( p ) = ψ ± (0 , p ) . It is perhaps worth noting that we have˜ ω = (cid:40) ω + + d ¯ a ∧ λ + + (¯ a − dλ + on { − (cid:15) < ¯ a ≤ } ω − + d ¯ a ∧ λ − + (¯ a + 1) dλ − on {− ≤ ¯ a < (cid:15) − } (6)Next we aim to define the two-form ¯ ω . To that end, we first define the desiredsmooth function β : R → R which satisfies the following conditions:(1) β (¯ a ) = 0 for | ¯ a | ≥ β (cid:48) (¯ a ) > − (cid:15) < | ¯ a | < β (¯ a ) = ¯ a − − (cid:15) < ¯ a < − (cid:15) (4) β (¯ a ) = ¯ a + 1 for (cid:15) − < ¯ a < (cid:15) − ω on { − (cid:15) < | ¯ a |} by the following:¯ ω = ω ± + d (cid:0) β (¯ a ) λ ± (cid:1) = ω ± + β (cid:48) (¯ a ) d ¯ a ∧ λ ± + β (¯ a ) dλ ± = β (cid:48) (¯ a ) d ¯ a ∧ λ ± + ˆ ω ± for ˆ ω ± = ω ± + β (¯ a ) dλ ± . (7)As a consequence of the definition of β together with ˜ ω expressed as in equation(6), we see that on { − (cid:15) < | ¯ a | < − (cid:15) } we have ˜ ω = ¯ ω , and hence we extendthe definition of ¯ ω so that ¯ ω (cid:12)(cid:12) {| ¯ a |≤ − (cid:15) } := (cid:101) ω , which yields a smooth two-form ¯ ω defined on all of W . This is the desired ¯ ω and establishes property (3b). It alsoimmediately follows that on {| ¯ a | < } the two-form ¯ ω is non-degenerate whichestablishes property (1a). This latter fact is established by observing that β × Id : (1 − (cid:15), × M + → ( − (cid:15), × M + β × Id : ( − , (cid:15) − × M − → (0 , (cid:15) ) × M − are diffeomorphisms which pull back ˜ ω to ¯ ω .At this point, we are prepared to define the smooth map Ψ : W → (cid:102) W . As a firststep, we define the smooth mapΨ : W + (cid:15) ∪ W − (cid:15) → (cid:102) W Ψ = ψ ± ◦ ( β × Id ) ◦ I − ± We then observe that on the collar regions W ± (cid:15) \ W ± (cid:15) we have I ◦ Ψ = Id , so thefollowing extension yields the desired smooth surjection.Ψ : W → (cid:102) W Ψ : = (cid:40) ψ ± ◦ ( β × Id ) ◦ I − ± (¯ q ) if | ¯ a (¯ q ) | > − (cid:15)I − (¯ q ) if | ¯ a (¯ q ) | < − (cid:15). From this, and our above constructions, we can immediately see that Ψ satisfiesthe desired properties, including Ψ ∗ ˜ w = ¯ ω .We now turn our attention to defining the almost complex structure J on W .To that end, we first observe that on the ends W ± (cid:15) we have the splitting T W = ker ( d ¯ a ∧ λ ± ) ⊕ ker ˆ ω ± (8)where ˆ ω ± is defined in equation (7), and we observe that by construction we have¯ ω (cid:12)(cid:12) ker ( d ¯ a ∧ λ ± ) = ˆ ω ± . We also define the vector field X ± by d ¯ a ( X ± ) = 0 λ ± ( X ± ) = 1 i X ± ˆ ω ± = 0 . From this we define J so that the following conditions hold J : ker ( d ¯ a ∧ λ ± ) → ker ( d ¯ a ∧ λ ± ) and J : ker ˆ ω ± → ker ˆ ω ± , and moreover, J∂ ¯ a = X and ˆ ω ± ( · , J · ) (cid:12)(cid:12) ker ( d ¯ a ∧ λ ± ) is symmetric and positive definite.Now, recall that on { − (cid:15) < | ¯ a |} we have ¯ ω = β (cid:48) (¯ a ) d ¯ a ∧ λ ± + ˆ ω ± , so that this J is¯ ω -compatible on { − (cid:15) < | ¯ a | < } , and because ¯ ω is non-degenerate on {| ¯ a | < } we may then smoothly extend J to be an almost complex structure which is ¯ ω -compatible on {| ¯ a | < } . This defines the desired J ; in particular, this establishesproperties (1b), (3c), and (4)At this point, we note that it only remains to define the Riemannian metric ¯ g ,show that J is a ¯ g -isometry, and properties (2) and (3d).To that end, we must first define the metric ¯ g , and to do that, we will first fix asmooth function χ which satisfies χ : R → [0 , χ (¯ a ) = (cid:40) | ¯ a | − < − (cid:15) | ¯ a | − > − (cid:15). Then, working on the ends, we define¯ g = (cid:16) θ (¯ a )( d ¯ a ∧ λ ± ) + ˆ ω ± (cid:17) ( · , J · )where θ (¯ a ) = χ (¯ a ) + (cid:0) − χ (¯ a ) (cid:1) β (cid:48) (¯ a ) and ˆ ω ± = ω ± + β (¯ a ) dλ ± . We see immediately from this definition that on the set {| ¯ a | − > − (cid:15) } that ¯ g is aRiemannian metric for which J is an isometry. Moreover, on the region { − (cid:15) < | ¯ a | < − (cid:15) } we have χ = 0 and hence on this region we have¯ g = (cid:16) β (cid:48) (¯ a )( d ¯ a ∧ λ ± ) + ω ± + β (¯ a ) dλ ± (cid:17) ( · , J · )= ¯ ω ( · , J · ) ERAL CURVES AND MINIMAL SETS 29 so that we may smoothly extend ¯ g by requiring that on {| a | < − (cid:15) } , we have¯ g = ¯ ω ( · , J · ). Consequently, ¯ g is indeed a Riemannian metric on W , for which J isalways an isometry. This, in turn, establishes properties (2) and (3d), which thencompletes the proof of Lemma 2.16. (cid:3) Remark 2.17 (adjustment of J ) . Recall (see for example Proposition 2.63 of [29] ) that if (cid:102) W is a finite dimensionalmanifold, and (cid:101) E → (cid:102) W is a rank n bundle with a symplectic bilinear form ˜ ω , thenon (cid:101) E there exists an almost complex structure (cid:101) J compatible with ˜ ω , and moreoverthe space of such almost complex structures is contractible. As a consequence, ifa symplectic cobordism ( (cid:102) W , ˜ ω ) is equipped with an almost complex structure (cid:101) J ,and K ⊂ (cid:102) W \ ∂ (cid:102) W is a compact set, then after shrinking (cid:15) > if necessary, theassociated extended symplectic cobordism W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) can be arrangedso that there exists a neighborhood O of K such that on Ψ − ( O ) we have Ψ ∗ (cid:101) J = J . Pseudoholomorphic Curves.
We now turn to pseudoholomorphic mapsand properties thereof.
Definition 2.18 (pseudoholomorphic map) . Let ( S, j ) and ( W, J ) be smooth almost complex manifolds with dim( S ) = 2 , eachpossibly with boundary. A C ∞ -smooth map u : S → W is said to be pseudoholomor-phic provided J · T u = T u · j . That is, the tangent map of u intertwines the almostcomplex structures on domain and target. Unless otherwise specified, we allow S tobe disconnected.We say such a map is proper provided the preimage of any compact set is com-pact. We say such a map is boundary-immersed provided either u : ∂S → W is animmersion, or else if ∂S = ∅ . Given a proper pseudoholomorphic map u : S → W , it will be convenient todenote the set of critical points by Z u := { ζ ∈ S : T ζ u = 0 } . Recall that anyconnected component S of S for which Z u ∩ S contains an interior accumulationpoint, we must have u (cid:12)(cid:12) S ≡ p ∈ W ; for details, see Lemma 2.4.1 in [30]. As such,for a pseudoholomorphic map u : S → W , the restriction of u to each connectedcomponent S ⊂ S is either a constant map, or else it is generally immersed in thefollowing sense. Definition 2.19 (generally immersed) . We say a pseudoholomorphic map is generally immersed provided the set of criticalpoints has no interior accumulation points.
Definition 2.20 (marked nodal Riemann surface) . A nodal Riemann surface is a triple ( S, j, D ) , with entries as follows. The firstentry, S , is a real two-dimensional manifold, which may have smooth boundary,but we require that each connected component of ∂S is compact. The second entry, j , is a smooth almost complex structure on S . Finally, the third entry, D ⊂ S \ ∂S ,is an unordered closed discrete set of pairs D = { d , d , d , d , . . . } which we callnodal points, and the pairs { d i , d i } we call nodal pairs.A marked nodal Riemann surface is the four-tuple ( S, j, µ, D ) where ( S, j, D ) isa nodal Riemann surface, and where µ ⊂ S \ ( D ∪ ∂S ) is a discrete closed set ofpoints. Remark 2.21 (nodal notation) . A careful reader may notice that in a nodal Riemann surface, the structure whichdetermines which nodal points are paired with which other nodal points to form anodal pair is implied by the notation but not explicitly provided in the tuple ( S, j, D ) .Although this ambiguity is standard in the literature, it can be made precise by letting D = { d , d , . . . } be a closed discrete set of points, and letting ι : D → D denotean involution which sends each nodal point d ∈ D to the unique point d (cid:48) ∈ D (with d (cid:54) = d (cid:48) ) with the property that { d, d (cid:48) } is a nodal pair. A nodal Riemann surface wouldthen be given by the tuple ( S, j, D, ι ) . In this way, ι ( d i ) = d i and ι ( d i ) = d i . Hereand throughout, we shall follow the more ambiguous but less cumbersome notationof Definition 2.20, and leave the obvious precisification to the reader. Associated to a nodal Riemann surface is the topological space | S | defined byidentifying a nodal point with the other point in its nodal pair; in other words, thespace S/ ( d i ∼ d i ).As in Section 4.4 of [4], we define S D to be the oriented blow-up of S at the points D , and we let Γ i := (cid:0) T d i ( S ) \ { } (cid:1) / R ∗ + ⊂ S D and Γ i := (cid:0) T d i ( S ) \ { } (cid:1) / R ∗ + ⊂ S D denote the newly created boundary circles over the d i . Definition 2.22 (decorated marked nodal Riemann surface) . A decorated marked nodal Riemann surface is a tuple ( S, j, µ, D, r ) where ( S, j, µ, D ) is a marked nodal Riemann surface, and r is a set of orientation reversing orthog-onal maps ¯ r ν : Γ ν → Γ ν and r ν : Γ ν → Γ ν , which we call decorations ; hereby orthogonal orientation reversing, we mean that r ν ( e iθ z ) = e − iθ r ν ( z ) for each z ∈ Γ ν . We also define S D,r to be the smooth surface obtained by gluing the com-ponents of S D along the boundary circles { Γ , Γ , Γ , Γ , . . . } via the decorations ¯ r ν and r ν . We will let Γ ν denote the special circles Γ ν = Γ ν ⊂ S D,r . We will also need the following definition.
Definition 2.23 (arithmetic genus) . Let S = ( S, j, µ, D ) be a marked nodal nodal Riemann surface. As above, let S D bethe oriented blow-up of S at the points D , and let S D,r denote the surface obtainedby gluing S D together along pairs of circles associated to pairs of nodal points. Wedefine the arithmetic genus of S to be the genus of S D,r . That is,
Genus arith ( S ) = Genus( S D,r ) . We note that it is more standard to define the arithmetic genus in terms of aformula involving the genera of connected components, number of marked points,number of nodal points, etc. It will be convenient for later applications to havethe above definition at our disposal, however it is equivalent to the more standardformulaic definition; see the Appendix of [13] for details.
Definition 2.24 (stable Riemann surface) . We say a compact marked nodal Riemann surface, ( S, j, µ, D ) , is stable if and onlyif for each connected component (cid:101) S ⊂ S we have χ ( (cid:101) S ) − (cid:101) S ∩ ( µ ∪ D )) < . Lemma 2.25 (uniformization) . Let ( S, j, µ, D ) be a stable compact marked nodal Riemann surface, possibly withboundary. Then there exists a unique smooth geodesically complete metric h on ERAL CURVES AND MINIMAL SETS 31 ˙ S := S \ ( µ ∪ D ) in the conformal class of j such that Area h ( ˙ S ) < ∞ , the Gausscurvature of h is identically − , and the boundary components of S are all h -geodesics.Proof. This is the well known uniformization theorem. A proof via variationalpartial differential equation methods in the case that µ ∪ D = ∅ = ∂S case can befound in [37]. The case with boundary can be treated by modifying the argumentin [37] to consider an associated Neumann boundary value problem. The case withpunctures can be treated by removing disks of arbitrarily small radius centered atpoints in Γ and taking limits. (cid:3) We call h the Poincar´e metric associated to ( S, j, µ, D ), and will often denoteit h j,µ ∪ D to denote the dependence upon both the conformal structure j and thespecial points µ ∪ D ; for example, see the notion of Gromov convergence given belowin Definition 2.35. Remark 2.26 (Orientations on Riemann surfaces) . Any Riemann surface is oriented by the almost complex structure so that ( v, jv ) is a positively oriented frame whenever v (cid:54) = 0 . Furthermore, if a Riemann surface ( S, j ) has boundary, then the boundary will be oriented by letting ν be an outwardpointing unit normal, and defining jν to be a positively oriented basis of ∂S . Definition 2.27 (Genus) . Let S be a two dimensional oriented manifold, possibly with boundary, with at mostcountably many connected components, and with the property that each connectedcomponent of ∂S is compact. Then(1) If S is closed and connected, then define Genus( S ) := g where χ ( S ) = 2 − g is the Euler characteristic of S .(2) If S is compact and connected with n boundary components, define (cid:101) S = (cid:0) S (cid:116) ( (cid:116) nk =1 D ) (cid:1) / ∼ to be the closed surface capped off by n disks, anddefine Genus( S ) := Genus( (cid:101) S ) .(3) If S is compact (possibly with boundary), then Genus( S ) is defined to be thesum of the genera of each connected component.(4) If S is not compact, then Genus( S ) is defined by taking any nested sequence S ⊂ S ⊂ S ⊂ · · · of compact surfaces (possibly with boundary) such that S k ⊂ S for all k ∈ N and such that S = ∪ k S k ; then we define Genus( S ) :=lim k →∞ Genus( S k ) . Remark 2.28 (Genus monotonicity) . Note that for compact surfaces with boundary,
Genus( · ) , thought of as a function,satisfies a notion of super-additivity made precise in Lemma 2.29 below. As aconsequence of this lemma, it immediately follows that if S (cid:48) and S (cid:48)(cid:48) are compactsurfaces with boundary and satisfy S (cid:48) ⊂ S (cid:48)(cid:48) , then Genus( S (cid:48) ) ≤ Genus( S (cid:48)(cid:48) ) , and hence for a non-compact surface S , Genus( S ) is well defined by Definition 2.27. Lemma 2.29 (Genus super-additivity) . Suppose S is a smooth compact oriented two-dimensional manifold, possibly withboundary. Suppose further that S = S ∪ S , for which the intersection S ∩ S
22 J.W. FISH AND H. HOFER consists of a finite union of pairwise disjoint smooth embedded loops for which S ∩ S = ( ∂S ) ∩ ( ∂S ) Then
Genus( S ∪ S ) ≥ Genus( S ) + Genus( S ) . Proof.
We begin by resolving a related problem. Indeed, suppose S (cid:48) is a smoothcompact oriented two-dimensional manifold, possibly with boundary, and furthersuppose there exists an smooth orientation reversing diffeomorphism from one con-nected component of ∂S (cid:48) to another. Define S := S (cid:48) / ∼ where x ∼ φ ( x ). Then weclaim(9) Genus( S ) ≥ Genus( S (cid:48) ) . To prove inequality (9), we recall the Euler characteristic of the surface S is givenby χ ( S ) = 2 π ( S ) − S ) − π ( ∂S ) , where π ( X ) denotes the number of connected components of the space X . How-ever, by the Gauss-Bonnet theorem we also have χ ( S ) = 12 π (cid:90) S K g dA + 12 π (cid:90) ∂S κ g ds where g is a Riemannian metric on S , K g is the Gaussian curvature of g , and κ g is the associated geodesic curvature. By choosing a metric on S (cid:48) for which the ∂S (cid:48) consists of geodesics and which smoothly descends to S , we see that χ ( S ) = χ ( S (cid:48) ) . Next we observe that π ( ∂S ) = π ( ∂S (cid:48) ) − π ( S ) = π ( S (cid:48) ) − e where e ∈ { , } . We now take the difference of the two following equations χ ( S ) = 2 π ( S ) − S ) − π ( ∂S ) χ ( S (cid:48) ) = 2 π ( S (cid:48) ) − S (cid:48) ) − π ( ∂S (cid:48) ) , and make use of the three above equations to find thatGenus( S ) − Genus( S (cid:48) ) = 1 − e ≥ . We conclude that Genus( S ) ≥ Genus( S (cid:48) ) . Next observe that if φ is a smooth orientation reversing diffeomorphism from theunion of several connected components of ∂S (cid:48) to the union of several other con-nected components of ∂S (cid:48) then again Genus( S ) ≥ Genus( S (cid:48) ) because the aboveargument can simply be iterated. However, Lemma 2.29 then follows immediatelyby letting S = S ∪ S and S (cid:48) = S (cid:116) S , since Genus( S (cid:116) S ) = Genus( S ) + Genus( S ) . This completes the proof of Lemma 2.29. (cid:3)
ERAL CURVES AND MINIMAL SETS 33
Definition 2.30 (marked nodal pseudoholomorphic curve) . A marked nodal pseudoholomorphic curve is a tuple u = ( u, S, j, W, J, µ, D ) withentries as follows. The triple ( S, j, µ, D ) is a marked nodal Riemann surface. Thepair ( W, J ) is a smooth real n -dimensional almost complex manifold, and u : S → W is a smooth map for which J · T u = T u · j . Finally, we require that u ( d i ) = u ( d i ) for each nodal pair { d i , d i } ⊂ D . Unless otherwise specified, we will allow S , the domain of a pseudoholomor-phic curve to be non-compact, to have smooth boundary, and to have unboundedtopology (i.e. countably infinite connected components, boundary components, andgenus). Definition 2.31 (stability and common types of pseudoholomorphic curves) . We will say that a pseudoholomorphic curve u = ( u, S, j, W, J, µ, D ) is(1) compact provided S is a compact manifold with smooth boundary,(2) closed provided S is a compact manifold without boundary,(3) connected provided that | S | is connected,(4) proper and boundary-immersed provided the map u : S → W is proper andboundary-immersed respectively.Lastly, we say that a boundary-immersed curve u is stable provided that for eachconnected component S ⊂ S on which the restriction u : S → W is constant, wehave (10) χ ( S ) − (cid:0) S ∩ ( µ ∪ D )) < . Remark 2.32 (on our notion of stability) . We begin by observing that our above definition of stability is notably different fromthe more precise definition established in Symplectic Field Theory, and instead hasmore in common with notion from Gromov-Witten theory. Indeed, in SFT the targetmanifold may be R × M with an R -action given by translation in the first factor,and therefore stable pseudoholomorphic curves (or buildings thereof ) are equivalenceclasses defined via this R -action. More specifically, a pseudoholomorphic building isstable only if it has the property that on each floor there is a connected componentwhich is not a trivial cylinder, and moreover that each constant component S satisfies the inequality of equation (10). In this way, an orbit cylinder by itselfis not stable in an SFT sense, but is stable in the sense of Definition 2.31 above.Because of this discrepancy, and because of the importance of the notion of stability,some discussion is warranted.To that end, we first recall that in the study of pseudoholomorphic curves, thenotion of stability arises predominantly so that a sequence of stable curves of fixedtopological type necessarily has a subsequence which has a unique stable limit.Moreover, the topology associated to this limit must be such that there exists agluing theorem which (at the very least) finds the tail of the subsequence given only atransverse limit curve. For a historical example, one can consider Gromov’s originaldefinition of compactness in [19] , which led to non-unique limits (due to arbitrarilycomplicated trees of constant spheres) and hence non-Hausdorff topologies on theassociated moduli spaces. This was then remedied by Kontsevich, who proposed thenotion of stability which yielded the desired unique limits. Similarly, in Symplectic For example, a sequence of connected curves of some specified genus and in a specific(relative) homology class.
Field Theory, unless one declares levels consisting only of trivial cylinders to beunstable (which is not done in Definition 2.31), then limit buildings obtained viacompactness will not be unique. We now explain why.The issue is that at present we do not have a full compactness theorem. Indeed,for the bulk of the argument below, we really only consider curves whose symplec-tization coordinate has an absolute minimum, and moreover if given a sequence ofsuch curves with suitable bounds (though not necessarily energy bounds) we extracta very weak notion of a limit curve. Indeed, in an SFT sense, what we find is onlythe bottom-most level in any naturally arising limit building. The reason for thisis two-fold. The first is simply to obtain a feral pseudoholomorphic curve (namelythe bottom-most level), which may have infinite energy, and we use this to find thedesired closed invariant subset of the Hamiltonian flow. This is rather analogous tohow Hofer first used a preliminary compactness/bubbling argument to establish theexistence of a finite energy plane in the symplectization of contact S , and henceproving that Reeb flows on S must have a periodic orbit; see [23] . Only laterwas a full compactness theorem established; see [4] . Consequently, stability for thepurposes of this manuscript need only take into account a single level of whateverbuilding structure the (eventually understood) full limit has.This raises the question: Given an understanding of SFT compactness, and anargument to extract a single level of the (supposed) feral limit building, why havewe not proved a full SFT compactness theorem for feral curves? Indeed, it is infact not difficult to build on the ideas here and in [13] to find many levels of a limitbuilding. However, at present there still remain several complications. The first isthat the limit may have infinitely many levels, each with positive ω -energy. Indeed,unlike feral planes, feral cylinders do not have an ω -energy threshold. Second, evenif one extracts all levels of a feral limit building which have positive ω -energy, it isnot yet clear if the sum of the ω -energy of each of the limit levels will equal the ω -energy of the curves in the approximating sequence. Without such knowledge, it isnot even clear the “full” limit has been found, since one should expect the ω -energyto be preserved in the compactification process. And third, even if a limit curveis understood which captures all the ω -energy there is to capture, at present it isnot understood how to glue two properly feral ends together. There are a variety ofspecial cases in which this seems possible, however it is not clear if these cases areexceptional or generic. As a consequence of all of these issues, one should regardthe notion of stability provided in Definition 2.31 as preliminary, proprietary, andrestricted to the specific needs of this manuscript. We expect an updated and moreprecise notion to naturally arise in future work. Definition 2.33 (decorated marked nodal pseudoholomorphic curve) . A decorated marked nodal pseudoholomorphic curve ( u , r ) is a pair for which u =( u, S, j, W, J, µ, D ) is a marked nodal pseudoholomorphic curve, and ( S, j, µ, D, r ) is a decorated marked nodal Riemann surface as in Definition 2.22. As above, welet S D,r be the smooth surface obtained by taking the oriented blow up of S at thepoints in D and then gluing the components of the result together along the boundarycircles Γ ν and Γ ν . Consequently, see that the smooth map u : S → W then lifts toa continuous map u : S D,r → W . Definition 2.34 (area of pseudoholomorphic curves) . Let u = ( u, S, j, W, J, µ, D ) be a marked nodal pseudoholomorphic curve which isproper and boundary immersed. Assume further that the almost Hermitian manifold ERAL CURVES AND MINIMAL SETS 35 ( W, J, g ) has no boundary. Let S const ⊂ S denote the union of connected componentsof S on which u is a constant map. As noted in the discussion following Definition2.18 (pseudoholomorphic map), the map u : S \ S const → W is generally immersed inthe sense of Definition 2.19. Consequently on S \ S const we can define the followingmetric dist u ∗ g ( ζ , ζ ) := inf (cid:110)(cid:82) (cid:104) ˙ γ ( t ) , ˙ γ ( t ) (cid:105) u ∗ g dt : γ ∈ C (cid:0) [0 , , S (cid:1) and γ ( i ) = ζ i (cid:111) , where our convention will be that if ζ and ζ lie in different connected components,then dist u ∗ g ( ζ , ζ ) := ∞ . Thus we may regard ( S \ S const , dist u ∗ g ) as a metric space,in which case it can be equipped with Hausdorff measures d H k . Note that if O ⊂ S \ S const is an open set on which u is an immersion, then d H ( O ) = Area u ∗ g ( O ) .As such, our convention will be to simply define the area of an arbitrary open set U ⊂ S \ S const to be Area u ∗ g ( U ) := d H ( U ) . Finally, for an arbitrary open set U ⊂ S we define Area u ∗ g ( U ) := d H ( U \ S const ) . Again, in the absence of a symplectic form, the above definition may seem foreign,so we pause for a moment to show that in the perhaps more familiar setup in whichΩ is a symplectic form, and J is an Ω-compatible almost complex structure so that g := Ω ◦ (Id × J ) is a Riemannian metric, the above definition of metric area ofa pseudoholomorphic curve agrees with the symplectic area as expected. To thatend, we suppose u : ( S, j ) → ( W, J ) is pseudoholomorphic map, and z ∈ S forwhich T z u (cid:54) = 0. We then let O ( z ) ⊂ S be an open neighborhood of z on which u is an immersion and on which there exist conformal coordinates ( s, t ) for which j∂ s = ∂ t . Consequently Ju s = u t , andΩ( u s , u t ) = (cid:107) u s (cid:107) g = (cid:107) Ju t (cid:107) g = (cid:107) u t (cid:107) g and (cid:104) u s , u t (cid:105) = Ω( u s , Ju t ) = − Ω( u s , u s ) = 0 , from which it follows that d H ( O ) = Area u ∗ g ( O )= (cid:90) O (cid:0) (cid:107) u s (cid:107) g (cid:107) u t (cid:107) g − (cid:104) u s , u t (cid:105) g (cid:1) ds ∧ dt = (cid:90) O (cid:107) u s (cid:107) g ds ∧ dt = (cid:90) O u ∗ Ω . We conclude that indeed, in the case that J is Ω-compatible, metric area as definedabove agree with symplectic area of pseudoholomorphic curves.We now turn our attention to issues of convergence of pseudoholomorphic curves.In what follows it will be important to recall that given a compact stable markednodal Riemann surface ( S, j, µ, D ), the associated Poincar´e metric, as provided inLemma 2.25, on S \ ( µ ∪ D ) is denoted by h j,µ ∪ D . We begin with our principlenotion of convergence of pseudoholomorphic curves. Definition 2.35 (Gromov convergence) . A sequence u k = ( u k , S k , j k , W, J k , µ k , D k ) of compact marked nodal stable boundary-immersed pseudoholomorphic curves is said to converge in a Gromov-sense to a compact marked nodal stable boundary-immersed pseudoholomorphic curve u =( u, S, j, W, J, µ, D ) provided the following are true for all sufficiently large k ∈ N .(1) J k → J in C ∞ .(2) There exist sets of marked points µ (cid:48) k ⊂ S k \ ( ∂S k ∪ µ k ∪ D k ) and µ (cid:48) ⊂ S \ ( ∂S ∪ µ ∪ D ) with the property that µ (cid:48) = µ (cid:48) k , and with the property that for eachconnected component (cid:101) S k of S k we have χ ( (cid:101) S k ) − (cid:0) (cid:101) S k ∩ ( µ k ∪ µ (cid:48) k ∪ D k ) (cid:1) < and for each connected component (cid:101) S of S we have χ ( (cid:101) S ) − (cid:0) (cid:101) S ∩ ( µ ∪ µ (cid:48) ∪ D ) (cid:1) < . (3) There exists a decoration r for u , a sequence of decorations r k for the u k , and sequences of diffeomorphisms φ k : S D,r → S D k ,r k k such that thefollowing hold(a) φ k ( µ ) = µ k (b) φ k ( µ (cid:48) ) = µ (cid:48) k (c) for each i = 1 , . . . , δ the curve φ k (Γ i ) is a h j k ,µ k ∪ µ (cid:48) k ∪ D k -geodesic in thepunctured surface S (cid:48) k := S k \ ( µ k ∪ µ (cid:48) k ∪ D k ) .(4) φ ∗ k h j k ,µ k ∪ µ (cid:48) k ∪ D k → h j,µ ∪ µ (cid:48) ∪ D in C ∞ loc (cid:0) S D,r \ ( µ ∪ µ (cid:48) ∪ i Γ i ) (cid:1) ; here we haveabused notation by letting h j,µ ∪ µ (cid:48) ∪ D also denote its lift to S D,r .(5) φ ∗ k u k → u in C ( S D,r ) .(6) φ ∗ k u k → u in C ∞ loc ( S D,r \ ∪ i Γ i ) .(7) For each connected component Λ of ∂S , the φ ∗ k h j k ,µ k ∪ µ (cid:48) k ∪ D k -length of Λ isuniformly bounded away from and ∞ . With this notion of convergence established, we can now provide the target-localized version of Gromov’s compactness theorem for pseudoholomorphic curves.
Theorem 2.36 (Target-local Gromov compactness) . Let ( W, J, g ) be an almost Hermitian manifold, possibly with boundary, and let ( J k , g k ) be a sequence of almost Hermitian structures which converge in C ∞ to ( J, g ) . Also let K , K ⊂ Int( W ) be compact regions, satisfying K ⊂ Int( K ) , andlet u k = ( u k , S k , j k , W, J k , µ k , D k ) be a sequence of stable compact marked nodalpseudoholomorphic curves satisfying u k ( ∂S k ) ∩ K = ∅ and suppose there exists alarge positive constant C > for which(1) Area u ∗ k g k ( S k ) ≤ C ,(2) Genus( S k ) ≤ C ,(3) (cid:0) µ k ∪ D k (cid:1) ≤ C Then, after passing to a subsequence (still denoted with subscripts k ), there existcompact surfaces with boundary (cid:101) S k ⊂ S k with the following properties(1) the following are compact pseudoholomorphic curves ( u k , (cid:101) S k , j k , µ k ∩ (cid:101) S k , D k ∩ (cid:101) S k ) (2) these domain-restricted converge in a Gromov sense to a compact stablemarked nodal boundary immersed pseudoholomorphic curve.(3) u k ( S k \ (cid:101) S k ) ⊂ W \ K . ERAL CURVES AND MINIMAL SETS 37
Proof.
This is essentially a restatement of Corollary 3.1 from [13], and a slightgeneralization of Theorem 3.1 from [12]. (cid:3)
Our final task of this section is to provide a notion of Gromov compactnessfor pseudoholomorphic curves in an exhaustive sense. The idea is best illustratedwith an example. Consider R × M equipped with an almost Hermitian structure( J, g ). Now let D denote the compact unit disk in C , and consider a sequence ofpseudoholomorphic curves u k : D → R × M with the following properties.(1) inf { a ◦ u k ( D ) } = 0 for each k ∈ N (2) sup { a ◦ u k ( D ) } = k = a ◦ u k ( ∂ D ) for each k ∈ N (3) Area u ∗ k g (cid:0) ( a ◦ u k ) − ([ − n, n ]) (cid:1) ≤ C n for each k ∈ N and each n ∈ N .Geometrically then, we have a sequence of disks, with a minimum in { } × M ,both a maximum and boundary in { k } × M , and a sort of locally bounded area.The question then becomes: Is there a notion of convergence for such curves whichcan be guaranteed after passing to a subsequence, and which yields a proper curvewithout boundary in R × M ? As it turns out, the answer is yes, and we make thenotion and the result precise with Definition 2.38 and Theorem 2.39 respectivelybelow. First however, we will need the following definition so that the desiredexhaustive compactness result can be stated in sufficient generality. Definition 2.37 (properly exhausting regions) . Let ( W , J, ¯ g ) be an almost Hermitian manifold, which need not be compact. We saya sequence of almost Hermitian manifolds ( W k , J k , g k ) properly exhaust ( W , J, ¯ g ) provided the following hold.(1) For each k ∈ N we have W k ⊂ W k +1 , and moreover W k is an open subsetof W k +1 in the W k +1 topology.(2) W = (cid:83) k ∈ N W k (3) The smooth structure on W k equals the smooth structure induced from W k +1 .(4) The set cl( W k ) ⊂ W k +1 is a compact manifold with smooth boundary.(5) Regarding ( J k , g k ) as almost Hermitian structures on W , we require ( J k , g k ) → ( J, ¯ g ) in C ∞ loc . Definition 2.38 (convergence in an exhaustive Gromov sense) . Let ( W , J, ¯ g ) be a smooth almost Hermitian manifold, not necessarily compact,and let ( W k , J k , g k ) be a sequence which properly exhausts ( W , J, ¯ g ) , in the senseof Definition 2.37. Suppose further that the tuples ¯ u = (¯ u, S, ¯ j, W , J, ¯ µ, D ) and,for each k ∈ N , u k = ( u k , S k , j k , W k , J k , µ k , D k ) , are each marked nodal properstable pseudoholomorphic curves without boundary. We say the sequence { u k } k ∈ N converges to ¯ u in an exhaustive Gromov sense provided there exists a collectionof compact smooth two dimensional manifolds with boundary { S (cid:96) } (cid:96) ∈ N with S (cid:96) ⊂ S for each (cid:96) ∈ N , and there exists a collection of compact smooth two dimensionalmanifolds with boundary { S (cid:96)k } (cid:96) ∈ N k ≥ (cid:96) with S (cid:96)k ⊂ S k for all k, (cid:96) ∈ N with k ≥ (cid:96) for whichthe following hold.(1) S (cid:96) ⊂ S (cid:96) +1 \ ∂S (cid:96) +1 for all (cid:96) ∈ N (2) S = (cid:83) (cid:96) ∈ N S (cid:96) (3) for each fixed k ∈ N and each ≤ (cid:96) ≤ k − we have S (cid:96)k ⊂ S (cid:96) +1 k \ ∂S (cid:96) +1 k (4) for each k ≥ (cid:96) ∈ N we have u − k ( W (cid:96) ) ⊂ S (cid:96)k , (5) for each fixed (cid:96) ∈ N , the sequence (cid:8)(cid:0) u k , S (cid:96)k , j k , W , J k , S (cid:96)k ∩ µ k , S (cid:96)k ∩ D k (cid:1)(cid:9) k ≥ (cid:96) is a sequence of compact marked nodal stable boundary-immersed pseu-doholomorphic curves which converges in a Gromov sense to the propermarked nodal stable boundary-immersed pseudoholomorphic curve (cid:0) ¯ u, S (cid:96) , ¯ j, W , J, S (cid:96) ∩ ¯ µ, S (cid:96) ∩ D (cid:1) . Theorem 2.39 (exhaustive Gromov compactness) . Let ( W , J, ¯ g ) be a smooth almost Hermitian manifold, not necessarily compact, andlet ( W k , J k , g k ) be a sequence which properly exhausts ( W , J, ¯ g ) , in the sense ofDefinition 2.37. Suppose further that the sequence denoted by { u k } k ∈ N = { ( u k , S k , j k , W k , J k , µ k , D k ) } k ∈ N is a sequence of proper stable marked nodal pseudoholomorphic curves withoutboundary for which there also exists a sequence of large constants C k with the prop-erty that for each fixed k ∈ N the following hold(C1) sup (cid:96) ≥ k Area u ∗ (cid:96) g (cid:96) ( (cid:98) S k(cid:96) ) ≤ C k (C2) sup (cid:96) ≥ k Genus( (cid:98) S k(cid:96) ) ≤ C k (C3) sup (cid:96) ≥ k (cid:0) ( µ (cid:96) ∪ D (cid:96) ) ∩ (cid:98) S k(cid:96) (cid:1) ≤ C k where (cid:98) S k(cid:96) := u − (cid:96) ( W k ) . Then a subsequence converges in an exhaustive Gromovsense to (¯ u, S, ¯ j, W , J, ¯ µ, D ) which is a proper stable marked nodal pseudoholomor-phic curve without boundary.Proof. This is a restatement of Theorem 1 from [13]. (cid:3) Existence of Minimal Subsets
The primary purpose of this section is to prove our main dynamical result,Theorem 1, as well as an almost immediate generalization, Theorem 2. It is usefulto note that substantial work is required to prove the first result, however thesecond result will follow from combining the foundational results about feral curvesdeveloped in later sections together with some well established techniques from [23].Due to the length of the proof of Theorem 1, we take a moment to sketch themain ideas in order to outline it. The steps are as follows.
Step 1:
Geometric and dynamical setup.
Here we embed our dynamical prob-lem into C P and build an extended symplectic cobordism from the empty set toa framed Hamiltonian manifold with dynamics conjugated to those on the givenenergy level. We also establish the existence of an embedded pseudoholomorphicsphere which we later show has some nice properties. Step 2:
Automatic transversality and an abundance of curves.
Here we definea moduli space of curves containing the previously found curve, and show thesecurves are very nice: they are each embedded, pairwise intersect exactly once, are
ERAL CURVES AND MINIMAL SETS 39 cut out transversely, and locally fill out an open set.
Step 3:
The moduli space M extends into the negative end of W . Here we showthat the curves in the moduli space above have images which descend arbitrarilydeep into the negative cylindrical end of the extended cobordism. Step 4:
An area estimate . Here we prove an area estimate, which can roughlybe regarded as showing that within a bounded distance from the non R -invariantregion the areas of the curves must be uniformly bounded. This estimate is notparticularly difficult to obtain, but it is necessary in order to apply Theorem 7. Step 5:
Trimming curves and applying the workhorse theorem.
Finally we use thearea bound from the previous step to trim away portions of the curve in the non R -invariant regions in the extended cobordism, and we show that the resulting familyof curves satisfy the hypotheses of Theorem 7, which then guarantees the existenceof a non-trivial closed invariant subset. The desired result is then immediate.3.1. Proof of Theorem 1.
We now prove our first main dynamical result regarding the non-minimality of theHamiltonian flow on compact hypersurfaces in R . Theorem 1 (First main dynamical result) . Consider R equipped with the standard symplectic structure and a Hamiltonian H ∈ C ∞ ( R , R ) for which M := H − (0) is a non-empty compact regular energylevel. Then the Hamiltonian flow on M is not minimal.Proof. Step 1:
Geometric and dynamical setup.
Here we build an extended symplectic cobordism, which has a negative end( M − , η − ) which is a framed Hamiltonian manifold with dynamics conjugated tothose on H − (0). We begin by letting ω denote the standard symplectic form on R given by ω = dx ∧ dy + dx ∧ dy . For each (cid:15) > L (cid:15) : R → R L (cid:15) ( x , y , x , y ) = (cid:16) x (cid:15) , y (cid:15) , x (cid:15) , y (cid:15) (cid:17) . We also define the smooth family of functions H (cid:15) : R → R by H (cid:15) := (cid:15) H ◦ L (cid:15) , andwe define manifolds M (cid:15) := H − (cid:15) (0). By definition, it follows that L (cid:15) : M (cid:15) → M is a diffeomorphism for each (cid:15) >
0. Moreover, along M and M (cid:15) respectively wehave X H ∈ Γ( T M ) and X H (cid:15) ∈ Γ( T M (cid:15) ). Further still, we have ω ( X H (cid:15) , v ) = − dH (cid:15) ( v )= − (cid:15) dH ( T L (cid:15) · v )= (cid:15) ω ( X H , T L (cid:15) · v ) = (cid:15) ω ( T L (cid:15) · ( T L (cid:15) ) − · X H , T L (cid:15) · v )= (cid:15) ( L ∗ (cid:15) ω ) (cid:0) L ∗ (cid:15) X H , v )= ω ( L ∗ (cid:15) X H , v ) , from which we see that X H (cid:15) = L ∗ (cid:15) X H . Consequently, the flow of the Hamiltonianvector field X H on M is not minimal if and only if the flow of the Hamiltonianvector field X H (cid:15) on M (cid:15) is not minimal.Next let ˜ ω denote the Fubini-Study metric on C P . Recall that there exists aholomorphic embedding ι : C → C P so that Σ := C P \ ι ( C ) is an embeddedcomplex submanifold holomorphically diffeomorphic to C P . Next, we regard R (cid:39) C so that ι : R → C P is an embedding. Then ι ∗ ˜ ω is a symplectic form on R ,so there exists a open neighborhood O of 0 ∈ R and a diffeomorphism φ : O → φ ( O ) ⊂ R for which φ ∗ ω = ι ∗ ˜ ω . Next, we fix (cid:15) > H − (cid:15) (0) ⊂ φ ( O ), and we define the open set U := { p ∈ R : H (cid:15) ( p ) < } ⊂ φ ( O ).However, we then have H − (cid:15) (0) = ∂ (cid:0) cl( U ) (cid:1) , and that φ ( O ) is an open neighborhoodof U ∪ H − (cid:15) (0) = cl( U ) and ι ◦ φ − : φ ( O ) → C P is an embedding for which ( ι ◦ φ − ) ∗ ˜ ω = ω . We then define the manifold (cid:102) W withsmooth boundary by (cid:102) W := C P \ ι ◦ φ − ( U ) and equip it with the symplectic form˜ ω . We then define the (not yet oriented) manifold M − := ∂ (cid:102) W , and equip it withthe two-form ω := i ∗ ˜ ω where i : M − (cid:44) → (cid:102) W is the canonical inclusion. Letting X := X H (cid:15) (cid:12)(cid:12) H − (cid:15) (0) , we see that by construction we have ω (cid:0) ( ι ◦ φ − ) ∗ X, · (cid:1) = 0and X never vanishes, so that we can define the one-form λ − via λ − ( X ) = 1 and ker λ − = T M − ∩ ( JT M − )where (cid:101) J ∈ Γ(End( T (cid:102) W )) is the (almost) complex structure induced on the real manifold (cid:102) W from multiplication by i = √− (cid:102) W ⊂ C P when regarded as a complex manifold. We note that (cid:101) J is ˜ ω -compatible in the sense that ˜ ω ( · , (cid:101) J · ) is aRiemannian metric. Defining η − := ( λ − , ω − ), we see that ( M − , η − ) is a framedHamiltonian manifold, with X η − = ( ι ◦ φ − ) ∗ X , and hence the flow of X η − on M − is minimal if and only if the flow of X H on H − (0) is minimal.We pause to sum up the salient features of our geometric construction. Givenour compact regular energy surface H − (0) ⊂ R inside ( R , ω ), we have con-structed a connected symplectic cobordism ( (cid:102) W , ˜ ω ), in the sense of Definition 2.11,from the empty framed Hamiltonian manifold to ( M − , η − ), where the flow of X η − is conjugated to that of X H on H − (0). Consequently, the flow of X H on H − (0) isminimal if and only if the flow of X η − is minimal on M − . Moreover, (cid:102) W \ ∂ (cid:102) W con-tains an embedded degree one sphere, (cid:101) Σ, with the property that its tangent planesare (cid:101) J invariant. Or in other words, there exists an embedded pseudoholomorphiccurve ˜ u = (˜ u, S , j, (cid:102) W , (cid:101) J, ∅ , ∅ ) ERAL CURVES AND MINIMAL SETS 41 with the property that ˜ u ( S ) = (cid:101) Σ ⊂ (cid:102) W ⊂ C P . Finally, we note that H ( C P , Z ) = Z and is generated by A (cid:101) Σ , where A (cid:101) Σ is the homology class associated to ˜ u . More-over, 1 π (cid:90) S ˜ u ∗ ˜ ω = 1 . At this point, we apply Lemma 2.16, which yields an associated symplectic cobor-dism W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ), in the sense of Definition 2.13, from the emptyframed Hamiltonian manifold to ( M − , η − ). Moreover, in light of Remark 2.17, wemay assume that ¯ u = (¯ u , S , j, W , J, ∅ , ∅ ) is a pseudoholomorphic curve, where¯ u := Ψ − ◦ ˜ u . We also define Σ = Ψ − ( (cid:102) W ) ⊂ W , and we let A Σ denote thehomology class associated to ¯ u . Step 2:
Automatic transversality and an abundance of curves.
Here we define moduli spaces of interest and establish properties thereof. Essen-tially we are interested in the moduli space of degree one spheres in W , and a certainpath connected component thereof, and we show these curves are embedded, cutout transversely, and pairwise intersect exactly once and do so transversely. Theseideas were all essentially introduced by Gromov in [19] and have been extensivelyemployed since, however we provide details for completeness, heavily referencingHofer-Lizan-Sikorav [24] and McDuff-Salamon [30] for detailed proofs.We begin by defining (cid:102) M to be the following set of pseudoholomorphic curves (cid:102) M = (cid:8) v = ( v, S , j, W , J, ∅ , ∅ ) : v is a pseudoholomorphic curve, and v and ¯ u are homologous. (cid:9) where here, as before, j denotes the standard (almost) complex structure on S .We equip (cid:102) M with the C ∞ topology, and we then let M denoted the path connectedcomponent of (cid:102) M which contains ¯ u .Recall that a closed pseudoholomorphic map u : ( S, j ) → ( W, J ) is a multi-ple cover provided there exists another Riemann surface ( S (cid:48) , j (cid:48) ), a holomorphicbranched covering φ : ( S, j ) → ( S (cid:48) , j (cid:48) ) with degree strictly greater than one, and apseudoholomorphic map u (cid:48) : ( S (cid:48) , j (cid:48) ) → ( W, J ) for which u = u (cid:48) ◦ φ . Also recall thata closed curve is said to be simple whenever it is not multiply covered. Proposition 3.1 (embeddedness and transvserse intersections) . Let W and (cid:102) M be as above. Then the following hold.(1) If u ∈ (cid:102) M then u : S → W is an embedding.(2) If u , v ∈ (cid:102) M with u ( S ) (cid:54) = v ( S ) then { ( ζ , ζ ) ∈ S × S : u ( ζ ) = v ( ζ ) } , and these intersections are transverse.Proof. As a first step, we recall Theorem 2.6.3 from [30] which is often called“positivity of intersections.” Roughly it states that if (
W, J ) is an almost com-plex four-manifold, and A , A ∈ H ( W ; Z ) are homology classes represented bysimple pseudoholomorphic curves ( u , S , j , W, J, ∅ , ∅ ) and ( u , S , j , W, J, ∅ , ∅ ) re-spectively, which have the property that there do not exist non-empty open sets U ⊂ S and U ⊂ S with the property that u ( U ) = u ( U ), then δ ( u , u ) ≤ A · A , (11)where δ ( u , u ) = { ( z , z ) ∈ S × S : u ( z ) = u ( z ) } . Moreover, we have equality in equation (11) if and only if the intersections aretransverse. From this, we may immediately draw the following conclusion: If u ∈ (cid:102) M , then u is simple. Indeed, if ¯ u represents the homology class A , and u ∈ (cid:102) M were not simple, then there would exist a homology class B represented by a simplepseudoholomorphic curve u (cid:48) which would necessarily satisfy0 < δ ( u (cid:48) , ¯ u ) ≤ B · A < A · A = 1 , which is impossible. As a consequence of this fact, together with unique continua-tion (see Theorem 2.3.2 and Corollary 2.3.3 in Section 2.3 of [30]), the second partof Proposition 3.1 follows immediately.To prove the first part of Proposition 3.1, we first recall Theorem 2.6.4 from[30], namely the adjunction inequality. It states that if ( W, J ) is an almost complexfour-manifold, and A ∈ H ( M ; Z ) is a homology class represented by a simplepseudoholomorphic curve u , then2 δ ( u ) − χ ( S ) ≤ A · A − c ( A )where δ ( u ) = { ( z , z ) ∈ S × S : z (cid:54) = z , u ( z ) = u ( z ) } . Moreover, equality holds if and only if u is an immersion with only transverseself-intersections. For curves u ∈ (cid:102) M we have χ ( S ) = χ ( S ) = 2, A · A = 1, and c ( A ) = 3. By definition we must have δ ( u ) ≥
0, and by the adjunction inequalitywe must then also have δ ( u ) ≤
0. It immediately follows that u is an embeddedpseudoholomorphic curve. This establishes the first part, and hence completes theproof of Proposition 3.1. (cid:3) As identified by Gromov in [19] and detailed by Hofer-Lizan-Sikorav in [24],for a given pseudoholomorphic curve u = ( u, S , j, W, J, ∅ , ∅ ), there exists a non-linear partial differential operator denoted ¯ ∂ ν called the normal Cauchy-Riemannoperator, which is defined on suitably small sections of the normal bundle over u ; that is, the sub-bundle of u ∗ T W consisting of those planes orthogonal to thetangent sub-bundle
T S ⊂ u ∗ T W . We call the space (cid:102) M / Aut( S ) the space of non-parameterized curves homologous to ¯ u , and note that a neighborhood of [ u ] ∈ (cid:102) M / Aut( S ) is given by the zero-set of the normal Cauchy-Riemann operator near u . The linearization of ¯ ∂ ν at u , denoted L ν , is a first order elliptic differentialoperator of the following form L ν u = ¯ ∂ + a, (12)where a ∈ Ω , (cid:0) End R ( ν u ) (cid:1) . In preparation for a later result, we now claim thefollowing. ERAL CURVES AND MINIMAL SETS 43
Lemma 3.2 (only one zero) . Let u ∈ (cid:102) M , and let L ν u be the linearization of ¯ ∂ ν at u as above. Let (cid:54) = σ ∈ ker( L ν u ) . Then { z ∈ S : σ ( z ) = 0 } . Proof.
First observe that c ( ν u ) = 1, where c ( ν u ) is the first Chern number ofthe normal bundle ν u over u ; that is, it is the algebraic count of zeros of a genericsection of ν . Consequently 1 ≤ { z ∈ S : σ ( z ) = 0 } . Next we claim the zeros of σ are isolated and each contributes positively to c ( ν u ).Indeed, this follows from the form L ν u takes, specifically equation (12), togetherwith the Carleman similarity principle. From this we conclude1 ≥ { z ∈ S : σ ( z ) = 0 } . The desired result is immediate. This completes the proof of Lemma 3.2. (cid:3)
Returning to our discussion of the linearized operator, we recall that L ν u isFredholm for suitable choices of Banach spaces; for example H¨older spaces C k,α orSobolev spaces W k,p with k ≥ p >
2. Moreover the index of L ν at a curve u = ( u, S, j, W, J, ∅ , ∅ ), is given byInd( L ν u ) = 2 (cid:0) c ( ν u ) + 1 − Genus( S ) (cid:1) where c ( ν u ) is the first Chern number of the normal bundle ν u over u , or equiva-lently if u is embedded, then c ( ν u ) = u · u , where u · u is the self-intersection number of u . One of the main results of [24] is thatif c ( ν u ) ≥ S ) − L ν u is surjective. As a consequence of Proposition3.1, each u ∈ (cid:102) M is embedded with Genus( S ) = 0 so that indeed c ( ν u ) = u · u = 1and hence L ν u is surjective with Ind( L ν ) = 4. By the implicit function theorem onBanach spaces it follows that a neighborhood of [ u ] ∈ (cid:102) M / Aut( S ) is a manifoldof dimension 4. Moreover, for each integer k ≥ O ⊂ ker( L ν u ) of the zero section and a smooth embedding E : O →C k,α ( ν u ) with the following properties.( F
1) For each σ ∈ O , E ( σ ) ∈ C ∞ ( ν u ); that is, E ( σ ) is a smooth section of thenormal bundle ν u over u .( F
2) The linearization of E at the zero section, denoted by T E , satisfies T E ( σ ) = σ for all σ ∈ ker( L ν u ).( F
3) For each σ ∈ O , there exists a pseudoholomorphic curve u σ = ( u σ , S , j, W , J, ∅ , ∅ )and a (not necessarily holomorphic) diffeomorphism ψ σ : S → S for which u σ ◦ ψ σ = exp (cid:0) E ( σ ) (cid:1) . See Theorem 2.3.5 of [30].
Moreover, for a continuous path [0 , → O denoted by τ (cid:55)→ σ τ , the ψ σ τ canbe found so that the map [0 , → C ∞ ( S , W ) given by τ (cid:55)→ u σ τ := exp u ◦ ψ − στ (cid:0) E ( σ τ ) (cid:1) is continuous.( F
4) The map F : O × S → WF ( σ, z ) = exp u ( z ) (cid:0) E ( σ ) (cid:1) is C ∞ smooth. Remark 3.3 (paths of reparametrizations) . To see the validity of the second part of property ( F { z , z , z } ⊂ S and σ ∈ O , the diffeomorphism ψ σ is uniquely determined by requiring ψ σ ( z i ) = z i for i ∈ { , , } . Given τ (cid:55)→ σ τ ,this then uniquely determines the map τ (cid:55)→ ψ σ τ , for our choice of z i . Continuityof the map τ (cid:55)→ u σ τ := exp u ◦ ψ − στ (cid:0) E ( σ τ ) (cid:1) then follows from the continuity of themap τ (cid:55)→ ψ σ τ , which essentially follows from Gromov compactness; here it may behelpful to recall that τ (cid:55)→ j τ := ( u σ τ ◦ ψ σ τ ) ∗ J is a continuous map into the space ofsmooth sections Γ (cid:0) End(
T S ) (cid:1) . With these facts recalled, we are now prepared to prove the following.
Lemma 3.4 (curve through nearby points) . Let W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) and (cid:102) M be as above, and let u ∈ (cid:102) M . Fix z ∈ S . Then there exists δ > with the property that for each q ∈ B δ (cid:0) u ( z ) (cid:1) , thereexists a continuous map h : [0 , → (cid:102) M for which h (0) = u , and h (1) = u =( u , S , j, W , J, ∅ , ∅ ) with q ∈ u ( S ) .Proof. We begin by letting V z be the fiber of ν u over the point z ∈ S . Next weclaim that for each point v ∈ V z there exists σ ∈ ker( L ν u ) such that σ ( z ) = v .To see this, suppose not. Then there exists a vector subspace Q ⊂ ker( L ν u ) ofdimension at least three for which σ ∈ Q implies σ ( z ) = 0. Because Q is at leastthree dimensional, it follows that there exists z ∈ S \ { z } and 0 (cid:54) = σ ∈ Q forwhich σ ( z ) = 0 = σ ( z ). However this contradicts Lemma 3.2. This contradictionthen establishes that indeed, for each point v ∈ V z , there exists σ ∈ ker( L ν u ) forwhich σ ( z ) = v .In light of this observation, we choose ˇ σ, ˆ σ ∈ O with O ⊂ ker( L ν u ) as above,so that V z = Span (cid:0) ˇ σ ( z ) , ˆ σ ( z ) (cid:1) . We then fix local coordinates ( s, t ) centered at z ∈ S in a neighborhood U ⊂ S and we then define the map F : U × D → W F ( s, t, x, y ) = exp u ( s,t ) (cid:0) E (cid:0) x ˇ σ + y ˆ σ ) (cid:1) where D = { ( x, y ) ∈ R : x + y < } . By property ( F
4) the map F is smooth.Moreover, by property ( F
2) the linearization T F (0 , , ,
0) is surjective. Conse-quently, there exists δ > B δ ( u ( z )) ⊂ F ( U × D ). Letting q ∈ B δ ( u ( z )),there exists ( s, t, x, y ) ∈ U × D so that F ( s, t, x, y ) = q . Define σ := x ˇ σ + y ˆ σ . Byconvexity of O , it follows that τ σ ∈ O for all τ ∈ [0 , F , → O given by τ (cid:55)→ σ τ := τ σ gives rise to a ERAL CURVES AND MINIMAL SETS 45 continuous map h : [0 , → (cid:102) M h ( τ ) = u σ τ = ( u σ τ , S , j, W , J, ∅ , ∅ )for which h (0) = u , and h (1) = u σ with q ∈ u σ ( S ). This completes the proof ofLemma 3.4. (cid:3) Step 3:
The moduli space M extends into the negative end of W .Here we aim to prove the following proposition. Proposition 3.5 (curves fall completely) . Let W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) and M be as above. Then for each a ≤ − , thereexists z ∈ S and u ∈ M for which a = a ◦ u ( z ) = inf z ∈ S a ◦ u ( z ) . In other words, the images of the curves in M extend as far down as we like into ( −∞ , − × M − ⊂ W .Proof. In order to proceed, we will need the following result.
Lemma 3.6 (bounded depth implies bounded area) . Let W = ( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) and M be as above. For each a ≤ − , there existsa C = C ( a , W , C ¯ u ) with the following property. For each u ∈ M for which u ( S ) ⊂ { p ∈ W : ¯ a ( p ) > a } , we have Area u ∗ ¯ g ( S ) ≤ C. Proof.
We begin by fixing δ so that − (cid:15) ≤ δ ≤ − (cid:15) so that δ is a regularvalue of a ◦ u . We then observe thatArea u ∗ ¯ g ( S ) = Area u ∗ ¯ g ( S + ) + Area u ∗ ¯ g ( S − )where S + = { ζ ∈ S : ¯ a ◦ u ( ζ ) ≥ δ } and S − = { ζ ∈ S : ¯ a ◦ u ( ζ ) ≤ δ } . Recall our definition of cylindrical end and core of W are adapted from equations(2) and (3) to our case in which M + = ∅ as follows:Core( W ) = { ¯ a ≥ − (cid:15) } Cyl − ( W ) = { ¯ a < − (cid:15) } . From this we immediately see that u ( S + ) ⊂ Core( W ) and u ( S − ) ⊂ Cyl − ( W ) . Letting C θ = C θ ( W ) denote the constant guaranteed by the final part of Remark2.15, we see immediately from equation (4) thatArea u ∗ g ( S + ) ≤ C θ (cid:90) S + u ∗ ¯ ω ≤ C θ (cid:90) S u ∗ ¯ ω ≤ C θ (cid:90) S ¯ u ∗ ¯ ω = πC θ . Somewhat similarly, we have u ( S − ) ⊂ Cyl − ( W ), and by Remark 2.15, we seethat Cyl − ( W ) has the structure of a realized Hamiltonian homotopy with suitablyadapted almost Hermitian structure. Consequently, Theorem 8 below guaranteesthe existence of a constant C A = C A ( a , W ) for whichArea u ∗ ¯ g ( S − ) ≤ C A . Combining these two inequalities yieldsArea u ∗ ¯ g ( S ) ≤ C A + C θ =: C, which completes the proof of Lemma 3.6. (cid:3) Before proceeding with the proof of Proposition 3.5, we state Theorem 8. Theproof is provided in Section 4.1.
Theorem 8 (area bounds in realized Hamiltonian homotopy) . Fix positive constants C H > , r > , and E > . Then there exists a constant C A = C A ( C H , r, E ) with the following significance. Let I × M, (ˆ λ, ˆ ω )) denote arealized Hamiltonian homotopy in the sense of Definition 2.9, and let ( J, g ) be anadapted almost Hermitian structure in the sense of Definition 2.10 with C H := sup q ∈I× M (cid:107) d ˆ λ q (cid:107) g ≤ C H . For each proper pseudoholomorphic map u : S → I r × M , where I r = ( a − r, a + r ) ⊂ I for which ∂S = ∅ , u − ( { a } × M ) = ∅ , and (cid:90) S u ∗ ω ≤ E < ∞ , the following also holds: Area u ∗ g ( S ) = (cid:90) S u ∗ ( da ∧ ˆ λ + ˆ ω ) ≤ C A . Additionally, for any [ a , a ] ⊂ I and any compact pseudoholomorphic map u : S → [ a , a ] × M for which a and a are regular values of a ◦ u and u − (cid:0) { a , a }× M (cid:1) = ∂S , the following also hold: (cid:90) Γ a u ∗ λ ≤ (cid:16) C H E + (cid:90) Γ a u ∗ λ (cid:17) e C H ( a − a ) , and (cid:90) Γ a u ∗ λ ≤ (cid:16) C H E + (cid:90) Γ a u ∗ λ (cid:17) e C H ( a − a ) , where Γ a i = ( a ◦ u ) − ( a i ) for i ∈ { , } . Similarly, for (cid:96) := min i ∈{ , } (cid:110) (cid:90) Γ ai u ∗ λ (cid:111) we have Area u ∗ g ( S ) ≤ ( C − H (cid:96) + E )( e C H ( a − a ) −
1) + E . With Lemma 3.6 established, we can now complete the proof of Proposition 3.5.Indeed, we do this by contradiction, and hence begin by assuming that Proposition3.5 is false. In this case, Lemma 3.6 guarantees that the curves in M have uniformlybounded area. We define the set U ⊂ W by the following: U = { ¯ q ∈ W : ∃ u ∈ M s.t. u ( z ) = ¯ q } . Claim 1:
The set U is open. ERAL CURVES AND MINIMAL SETS 47
This follows immediately from Lemma 3.4.
Claim 2:
The set U is closed.To see this, we take a sequence ¯ q k → ¯ q with { ¯ q k } k ∈ N ⊂ U . Then there exist u k ∈ M with ¯ q k ∈ u k ( S ), which have uniformly bounded area and genus. ByTheorem 2.36, a sub-sequence converges to the stable curve u = (cid:0) u, S, j, W , J, ∅ , D (cid:1) with ¯ q ∈ u ( S ) and S = (cid:116) n +1 i =1 S , for some n ≥ Case I: Σ ⊂ u ( S ).In this case there must exist a connected component S ⊂ S for which u ( S ) = Σ.If S = S , then D = ∅ , and hence u ∈ M , and we are done, so assume S (cid:54) = S .In this case, we denote any remaining connected components by S , . . . , S n . Bystability of u , we may re-order the S i so that u : S → W is not a constantmap and u ( S ) ∩ Σ (cid:54) = ∅ . However, in a neighborhood of Σ, the two-form ¯ ω issymplectic and evaluates positively on J -complex lines. Consequently (cid:82) S u ∗ ¯ ω > ω evaluates non-negatively on J -complex lines in general, it followsthat (cid:90) S u ∗ ¯ ω > (cid:90) S ¯ u ∗ ¯ ω which is impossible since u and ¯ u represent the same homology class. This con-tradiction establishes that S = S , and hence u ∈ M . Case II: Σ (cid:54)⊂ u ( S ).By positivity of intersections, there exists exactly one connected component S of S for which u ( S ) ∩ Σ (cid:54) = ∅ . As before, if S = S then we are done, so we considerthe case that S has other connected components which we denote S , . . . , S n . Nextwe note that because ¯ ω evaluates non-negatively on J -complex lines, it follows that (cid:90) S i u ∗ ¯ ω ≥ i ∈ { , . . . n } . As before, it follows from stability of u that we may reorder the S i so that u : S → W is non-constant. Observe that by unique continuation, we must have (cid:82) S u ∗ ¯ ω >
0. However, because ¯ ω = Ψ ∗ ˜ ω , and because (cid:82) S i u ∗ ¯ ω > i ∈ { , } , itfollows that (cid:82) S i (Ψ ◦ u ) ∗ ˜ ω ≥ i ∈ { , } , and hence1 = 1 π (cid:90) S u ∗ ¯ ω = n (cid:88) i =0 π (cid:90) S i u ∗ ¯ ω ≥ π (cid:90) S u ∗ ¯ ω + 1 π (cid:90) S u ∗ ¯ ω = 1 π (cid:90) S (Ψ ◦ u ) ∗ ˜ ω + 1 π (cid:90) S (Ψ ◦ u ) ∗ ˜ ω ≥ . . This contradiction establishes that S = S , and hence u ∈ M . This completesClaim 2. At this point we realize that U is both open and closed, and hencemust equal W , which is impossible. This contradiction then completes the proof ofProposition 3.5. (cid:3) Step 4:
An area estimate .Here we prove the following.
Lemma 3.7 (ad hoc area estimate) . There exists a C = C ( W ) > with the following significance. Let u ∈ M . Define (cid:101) S = ( a ◦ u ) − ( (cid:101) I ) where I is the interval (cid:101) I = ( − , − . Then Area u ∗ g ( (cid:101) S ) ≤ C. Proof.
For notational convenience, we define the interval I (cid:48) = (cid:0) (cid:15) − , (cid:15) − (cid:1) . Wethen define the region (cid:99) W = (cid:8) ¯ q ∈ W : ¯ a (¯ q ) ∈ I (cid:48) (cid:9) . We then recall that because W is an extended symplectic cobordism in the senseof Definition 2.13, it follows that on W we have¯ ω = (cid:16) β (cid:48) (¯ a )( d ¯ a ∧ λ − ) (cid:17) + (cid:16) ω − + β (¯ a ) dλ − (cid:17) where we define the positive constants c β and c (cid:48) β by0 < c β := inf ¯ a ∈I (cid:48) β (¯ a ) and 0 < c (cid:48) β := inf ¯ a ∈I (cid:48) β (cid:48) (¯ a ) . Because J preserves the kernel of each of d ¯ a ∧ λ − and ω − + βdλ − , and becauseeach of these two-forms evaluates non-negatively on J -complex lines, the followingholds: (cid:90) u − ( (cid:99) W ) u ∗ ( d ¯ a ∧ λ − ) ≤ c (cid:48)− β (cid:90) u − ( (cid:99) W ) u ∗ ( β (cid:48) d ¯ a ∧ λ − ) ≤ c (cid:48)− β (cid:90) u − ( (cid:99) W ) u ∗ ( β (cid:48) d ¯ a ∧ λ − ) + u ∗ ( ω − + βdλ − ) ≤ c (cid:48)− β (cid:90) u − ( (cid:99) W ) u ∗ ¯ ω ≤ c (cid:48)− β π. However, by the co-area formula we also have (cid:90) u − ( (cid:99) W ) u ∗ ( d ¯ a ∧ λ − ) = (cid:90) I (cid:48) (cid:16) (cid:90) Γ t u ∗ λ − (cid:17) dt, See Lemma 4.13 below for a precise statement and proof of the required version of the co-areaformula.
ERAL CURVES AND MINIMAL SETS 49 where Γ t = ( a ◦ u ) − ( t ) for each regular value t of a ◦ u . Combining the abovetwo observations then guarantees the existence a regular value t of a ◦ u satisfying (cid:15) − < t < (cid:15) −
1, and with the property that (cid:90) Γ t u ∗ λ − ≤ π(cid:15)c (cid:48) β . In particular, (cid:82) Γ t u ∗ λ − is bounded in terms of the geometry of W . However, wethen note that (cid:99) W ⊂ Cyl − ( W ), and by Remark 2.15 we recall that Cyl − ( W ) has thestructure of a realized Hamiltonian homotopy. Consequently Theorem 8 applies,which guarantees the existence of a constant C = C ( W ) > u ∗ g ( (cid:101) S ) ≤ Area u ∗ g (cid:0) ( a ◦ u ) − (cid:0) ( − , t ) (cid:1)(cid:1) ≤ C. This is the desired inequality which proves Lemma 3.7. (cid:3)
Step 5:
Trimming curves and applying the workhorse theorem .In order to complete the proof of Theorem 1, we will apply Theorem 7 to acollection of curves which we now construct from curves in M . The rough idea isto carefully trim curves from M so that we may regard the resulting compact curveswith boundary as having images in the translation invariant region of Cyl − ( W ) ina manner that Theorem 7 applies. After reviewing the hypotheses of Theorem 7,the main concern becomes how to trim the curves so the boundary of the domainshave images in ( − , − × M − ⊂ Cyl − ( W ), and so that the number of boundarycomponents stays bounded. To that end, we will need the following result. Lemma 3.8 (bounds on number of boundary components) . Let ( W, J, g ) be a compact almost Hermitian manifold with smooth boundary, andlet ( J k , g k ) be a sequence of almost Hermitian structures which converge in C ∞ ( W ) to ( J, g ) . Let I be an index set, possibly uncountable, and denote the interior of W by W := Int( W ) . Suppose there exists a constant C > , and a set of stable properpseudoholomorphic curves u k,ι = ( u k,ι , S k,ι , j k,ι , W , J k , µ k,ι , D k,ι ) which satisfy(1) Area u ∗ k,ι g k ( S k,ι ) < C (2) Genus( S k,ι ) < C (3) µ k,ι ∪ D k,ι ) ≤ C Then for each sufficiently small δ > , there exists another constant C (cid:48) = C (cid:48) ( δ ) > with the following property. For each ( k, ι ) ∈ N × I , there exists a compacttwo-dimensional submanifold (possibly with smooth boundary) (cid:101) S k,ι ⊂ S k,ι with theproperty that sup ζ ∈ S k,ι \ (cid:101) S k,ι dist g (cid:0) u k,ι ( ζ ) , ∂W (cid:1) ≤ δ and π ( ∂ (cid:101) S k,ι ) ≤ C (cid:48) ; here π ( X ) denotes the number of connected components of X . Proof.
We suppose the lemma is not true and aim to derive a contradiction. To thatend, there must exist a sequence (cid:96) (cid:55)→ ( k (cid:96) , ι (cid:96) ) with the property that for each compacttwo-dimensional submanifold (possibly with smooth boundary) (cid:101) S k (cid:96) ,ι (cid:96) ⊂ S k (cid:96) ,ι (cid:96) thatsatisfies sup ζ ∈ S k(cid:96),ι(cid:96) \ (cid:101) S k(cid:96),ι(cid:96) dist g (cid:0) u k (cid:96) ,ι (cid:96) ( ζ ) , ∂W (cid:1) ≤ δ also satisfies π ( ∂ (cid:101) S k (cid:96) ,ι (cid:96) ) ≥ (cid:96). Without loss of generality, we may assume that the map (cid:96) (cid:55)→ k (cid:96) is either strictlymonotonically increasing or else constant. Here we shall assume (cid:96) (cid:55)→ k (cid:96) is strictlymonotonic, and leave trivial modifications for the constant case to the reader. Next,for notational convenience, we define a sequence of pseudoholomorphic curves bythe following.(ˆ u (cid:96) , (cid:98) S (cid:96) , ˆ j (cid:96) , W , (cid:98) J (cid:96) , ˆ µ (cid:96) , (cid:98) D (cid:96) ) := ( u k (cid:96) ,ι (cid:96) , S k (cid:96) ,ι (cid:96) , j k (cid:96) ,ι (cid:96) , W , J k (cid:96) , µ k (cid:96) ,ι (cid:96) , D k (cid:96) ,ι (cid:96) )We now observe that by assumption, this sequence of pseudoholomorphic curvesis stable and proper in W , and (cid:98) J (cid:96) → J in C ∞ ( W ), and they have uniformlybounded area, genus, number of special points. We then define the compact set K := (cid:110) p ∈ W : dist g ( p, ∂W ) ≥ δ (cid:111) , and apply Theorem 2.36, which guarantees that after passing to a subsequence (stilldenoted with subscripts (cid:96) ), there exist compact manifolds (possibly with smoothboundary) denoted by (cid:101) S (cid:96) ⊂ (cid:98) S (cid:96) such thatˆ u (cid:96) ( (cid:98) S (cid:96) \ (cid:101) S (cid:96) ) ⊂ (cid:110) p ∈ W : dist g ( p, ∂W ) < δ (cid:111) and the curves (ˆ u (cid:96) , (cid:101) S (cid:96) , ˆ j (cid:96) , W , (cid:98) J (cid:96) , ˆ µ (cid:96) , (cid:98) D (cid:96) )converge in a Gromov sense. In particular, for all sufficiently large (cid:96) , we have π ( ∂ (cid:101) S (cid:96) ) = n for all sufficiently large (cid:96) . But since (cid:101) S (cid:96) ⊂ (cid:98) S (cid:96) = S k (cid:96) ,ι (cid:96) , we must have π ( ∂ (cid:101) S (cid:96) ) ≥ (cid:96) have the desired contradiction. This completes the proof of Lemma3.8 (cid:3) We are now prepared to complete the proof of Theorem 1. First, for each realnumber b <
0, we fix ˆ u b ∈ M such thatˆ u b = (cid:0) ˆ u b , S , ˆ j b , W , J, ∅ , ∅ (cid:1) , and inf z ∈ S a ◦ ˆ u b ( z ) = a ◦ ˆ u b ( z b ) = b − . Next, define the manifoldˇ W := (cid:8) ¯ q ∈ W : − ≤ ¯ a (¯ q ) ≤ − (cid:9) , its interior ˇ W := (cid:8) ¯ q ∈ W : − < ¯ a (¯ q ) < − (cid:9) , the surfaces ˇ S b = { z ∈ S : ˆ u b ( z ) ∈ ˇ W } , ERAL CURVES AND MINIMAL SETS 51 and the pseudoholomorphic curvesˇ u b = (ˇ u b , ˇ S b , ˇ j b , ˇ W, J, ∅ , ∅ ) , where ˇ u b = ˆ u b (cid:12)(cid:12) ˇ S b and ˇ j b = ˆ j b (cid:12)(cid:12) ˇ S b . We then apply Lemma 3.8 to the curves ˇ u b in( ˇ W, J, ¯ g ) with δ < to obtain the compact surfaces with boundary denoted: (cid:101) ˇ S b ⊂ ˇ S b ⊂ S . Finally, we define the compact surfaces with boundary denoted by S b to be theconnected component of (cid:101) ˇ S b ∪ ( u b ) − (cid:16) { ¯ q ∈ W : ¯ a (¯ q ) < − } (cid:17) which contains a point z b ∈ S so thatinf z ∈ S a ◦ ˆ u b ( z ) = a ◦ ˆ u b ( z b ) = b − . By construction, we then have that each S b is compact, connected, with ˆ u b ( ∂S b ) ⊂ ( − , − × M − ⊂ Cyl − ( W ), andsup b< π ( ∂S b ) < ∞ . Next, we let u bk = u b = (cid:0) u b , S b , j b , ( −∞ , × M − , J, ∅ , ∅ (cid:1) . We also require that j b = ˆ j b (cid:12)(cid:12) S b and define u b = Sh − ◦ (Φ − ) − ◦ ˆ u b , where Φ − : ( −∞ , − (cid:15) ) × M − → Cyl − ( W ) is the diffeomorphism guaranteed byRemark 2.15, and Sh − : R × M − → R × M − is the shift map given by Sh − ( a, p ) =( a + 2 , p ).With our curves u bk = u b defined, we now collect the properties they have.(P1) each S b = | S b | is connected(P2) u bk is compact and u b ( ∂S b ) ⊂ (0 , × M − ,(P3) inf ζ ∈ S b a ◦ u b ( ζ ) = b (P4) there exists a continuous path α : [0 , → | S b | = S b satisfying a ◦ u b ◦ α (0) = b and α (1) ∈ ∂S b (P5) Genus( S b ) = 0(P6) (cid:82) S b ( u b ) ∗ ω − ≤ π (P7) D b = 0(P8) The number of connected components of ∂S bk is uniformly boundedMoreover, by Proposition 3.1, it follows that for any b, b (cid:48) < b (cid:54) = b (cid:48) we have (cid:0) u b ( S b ) ∩ u b (cid:48) ( S b (cid:48) ) (cid:1) ≤ . From this we see that the hypotheses of Theorem 7 are satisfied, and hence weconclude the existence of a closed set Ξ ⊂ M − satisfying ∅ (cid:54) = Ξ (cid:54) = M − whichis invariant under the flow of the Hamiltonian vector field X η − . Recalling theHamiltonian H : R → R given in they hypotheses of Theorem 1, we note that byconstruction the Hamiltonian flow on M − is conjugated to the flow on on H − (0), and hence we conclude that the Hamiltonian flow on H − (0) is not minimal. Thiscompletes the proof of Theorem 1. (cid:3) Proof of Theorem 2.
We are now prepared to prove the second main dynamical result. We begin with afew preliminaries. In what follows, we let D denote the closed disk in the complexplane, given by D = { ( s, t ) ∈ R : s + t ≤ } . Definition 3.9 (contact type) . Let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) . We say ( M, η ) is contact type provided that ω = dλ . Definition 3.10 (tight/overtwisted) . Let ( M, η ) be a three-dimensional framed Hamiltonian manifold of contact type. Wesay ( M, η ) is overtwisted provided there exists an embedding φ : D → M so thatthe one form φ ∗ λ has { } ∪ ∂ D as its zero set. If no such embedding exists, thenwe call ( M, η ) tight . Theorem 2 (second main dynamical result) . Let ( M ± , η ± ) be a pair of three-dimensional framed Hamiltonian manifolds, andlet ( (cid:102) W , ˜ ω ) be a symplectic cobordism from ( M + , η + ) to ( M − , η − ) in the sense ofDefinition 2.11. Suppose that ˜ ω is exact, M − is connected, and that ( M + , η + ) iscontact type and has a connected component M (cid:48) which is either S , overtwisted, orthere exists an embedded S in M (cid:48) ⊂ ∂ (cid:102) W which is homotopically nontrivial in (cid:102) W .Then the flow of the Hamiltonian vector field X η − on M − is not minimal.Proof. We begin by noting that the core arguments of the proof of Theorem 2 areidentical to those in Theorem 1, with some minor modifications from [23]. As such,our argument here will be brief.The first key observation is to see that for an almost Hermitian structure adaptedto a framed Hamiltonian manifold which is contact type, the function a ◦ u : S → R × M has a maximum principle whenever u : S → R × M is a pseudoholomorphicmap. That is, a ◦ u can have no interior local maxima. To see this, observe that (cid:0) ∆( a ◦ u ) (cid:1) ds ∧ dt = (cid:0) ( a ◦ u ) ss + ( a ◦ u ) tt (cid:1) ds ∧ dt = d (cid:0) ( a ◦ u ) s dt − ( a ◦ u ) t ds (cid:1) = − d (cid:0) d ( a ◦ u ) ◦ j (cid:1) = − d (cid:0) da ( du ◦ j ) (cid:1) = − d (cid:0) da ( J ◦ du ) (cid:1) = d (cid:0) λ ( du ) (cid:1) = u ∗ dλ = u ∗ ω ≥ . Following [23], we then break the problem into three cases.
Case I:
The connected component ( M (cid:48) , λ + ) is tight S . ERAL CURVES AND MINIMAL SETS 53
In this case, it follows from deep work of Eliashberg (see [8] and [10]) that upto diffeomorphism there exists a unique positive tight contact structure on S , andmoreover there exists a smooth embedding φ : M (cid:48) → R for which λ + (cid:12)(cid:12) M (cid:48) = φ ∗ ( x dy + x dy ) . Additionally, φ ( M (cid:48) ) is the boundary of a compact star-shaped set O ⊂ R with O diffeomorphic to a compact four-ball.Following the construction in the proof of Theorem 1, it then becomes possible tobuild a symplectic cobordism obtained by symplectically capping off M (cid:48) ⊂ ∂ (cid:102) W by C P \ O . The resulting manifold, denoted ( ˇ W, ˇ ω ), is then a symplectic cobordismfrom ( M + \ M (cid:48) , η + ) to ( M − , η − ). One can then find an almost Hermitian structure( (cid:101) J, ˜ g ) on ˇ W for which (cid:101) J is adapted to ˇ ω and for which there exists an embeddedpseudoholomorphic sphere u : S → ˇ W which has the same properties as ¯ u fromthe proof of Theorem 1. That is, one considers the moduli space M of non-nodalcurves which are homotopic (through non-nodal pseudoholomorphic curves) to thisspecial curve. Curves in this moduli space are cut out transversely and pairwiseintersect at exactly one point. By the same means as in the proof Theorem 1, oneshows that if this family of curves is contained in a compact region, then the areais uniformly bounded.By positivity of intersections and exactness of ¯ ω , bubbling is impossible, andhence one can show that the set of points in the extension of ˇ W which are in theimage of a curve in M is both open and closed if the curves stay in a compact re-gion. This contradiction establishes that the curves must escape into a cylindricalend of the extended cobordism, but the maximum principle prevents them fromescaping into the positive end. Thus the curves must extend all the way down intothe single negative end of the extension of ˇ W , while each still intersects the initialcurve. Trimming the curves as in the proof of Theorem 1 then yields a sequenceof curves to which Theorem 7 applies, and hence the non-minimality of the flow of X η − on M − is established. Case II:
The manifold ( M (cid:48) , η + ) is overtwisted. This case relies more heavily on input from [23]. Begin by letting the tuple W =( W , ¯ ω, J, ¯ g, ¯ a, ∂ ¯ a , (cid:15) ) denote the extension associated to ( (cid:102) W , ˜ ω ). Letting φ : D → M (cid:48) be the overtwisted disk guaranteed to exist, we lift this to an embedding ˜ φ : D → R + × M (cid:48) via ˜ φ ( s, t ) = (cid:0) , φ ( s, t ) (cid:1) ∈ R × M (cid:48) . Letting Φ + : (1 − (cid:15), ∞ ) × M + → Cyl + ( W ) denote the embedding guaranteed by Remark 2.15, we then define theembedded disk Σ := Φ + ( ˜ φ ( D )) . Define the point e := Φ + ( ˜ φ (0)) ∈ Σ. Then, following [23], one constructs a familyof pseudoholomorphic curves of the form u : D → W with u : ∂ D →
Σso that u ( ∂ D ) is transverse to T Σ ∩ ker λ + and winds around e precisely once. Asis shown in [23], such curves are the zero set of a smooth non-linear Fredholmsection the linearization of which is always surjective. Additionally, the curves A compact set
K ⊂ R is said to be star-shaped provided that for each ( x , y , x , y ) ∈ K and each τ ∈ [0 , τx , τy , τx , τy ) ∈ K . are pairwise disjoint, and the images of their boundaries locally foliate Σ. Anadditional crucial fact is that the boundaries u ( ∂ D ) always stay transverse to thecharacteristic foliation given by the integral curves of T Σ ∩ ker λ + , and hence theboundaries u ( ∂ D ) must always stay disjoint from ∂ Σ.At this point, we follow the script from Case I and from the proof of Theorem 1.We let M denote the moduli space of such curves, and we note that if there exists acompact set K ⊂ W which contains the images of all the curves in M , then the setof points in Σ which are in the image of u (cid:12)(cid:12) ∂ D for u ∈ M is both open and closed,which is impossible. Here again we are making use of the fact that the existenceof such a K guarantees a uniform area bound as before, which then guaranteesGromov convergence, which establishes closedness. Again, the maximum principleprevents curves from escaping into Cyl + ( W ), so the curves must instead escapeinto Cyl − ( W ), and again the curves can be trimmed so that Theorem 2 applies,and again the flow of X η − on M − is not minimal. Case III:
There exists embedded S ⊂ M (cid:48) which is homotopically nontrivial in (cid:102) W . In this case we again follow [23] rather closely. In particular, by assumption, thereexists an embedded sphere in M (cid:48) which when included into (cid:102) W is homotopicallynontrivial. We also may assume that M (cid:48) is tight, since the overtwisted case hasalready been established. Consequently, we may perturb this sphere, keeping itembedded, so that there exist precisely two points { e + , e − } ⊂ Σ at which we have T Σ = ker λ + , and all integral curves of T Σ ∩ ker λ + have e + as one end point and e − as the other. As in Case II, we lift this sphere into a level set Σ ⊂ { } × M (cid:48) ⊂ W .As in the proof of Case II, we then construct a family of pseudoholomorphic diskswith boundary in the sphere Σ, each winding around e ± exactly once. These curvesagain have similar properties, like being cut out transversely, and they are pairwisedisjoint and have boundaries which locally foliate Σ. Once again one shows that ifthe images of the curves in this moduli space are contained in some compact set,then the set of points in Σ which are in the image of the boundaries of curves in thismoduli space is both open and closed in Σ, and hence are all of Σ. However, if thisis the case, then, as in [23], one can use the moduli space of curves to show that Σ ishomotopically trivial, which is impossible. Consequently, the images of the curvescannot stay in a compact region, so they must escape out into a cylindrical end of W , and by the maximum principle it cannot be the positive end. Again, curvesescape into the negative end, are pairwise disjoint, with a suitable area bound toobtain the appropriate trimmings to apply the workhorse theorem and again theflow of X η − on M − is not minimal. (cid:3) Supporting Proofs
In this section we prove the main foundational results about feral curves whichare needed to prove Theorem 1 and Theorem 2. Each of the following sections isdedicated to precisely one of the proofs of Theorem 3 through Theorem 7.4.1.
Proof of Theorem 3: Exponential Area Bounds.
The main purpose of this section is to prove Theorem 3 as well as several impor-tant generalizations. Our first step will be to give a brief overview of the structure
ERAL CURVES AND MINIMAL SETS 55 of the proofs, while indicating the methods used to overcome certain obstacles.Throughout this section, we will assume that M is a closed manifold equipped witha framed Hamiltonian structure η = ( λ, ω ), and that ( J, g ) is an η -adapted almostHermitian structure on R × M .4.1.1. The Rough Sketch.
Before proceeding into some of the technical (and tediousbut elementary) details, we first provide the core idea in a model scenario. After-wards, we describe how to generalize. To that end, we first need some definitions.We begin by assuming u : S → R × M is a pseudoholomorphic map, with thefollowing properties.(1) u ( S ) ⊂ [0 , r ] × M for some fixed positive r > ∂S = u − ( { , r } × M )(3) { ζ ∈ S : d ( a ◦ u ) ζ = 0 } = ∅ , where a is the symplectization coordinate on R × M .Geometrically then, we should think of S as an annulus (or a finite union of an-nuli), and u maps one boundary component to { } × M , and it maps the otherboundary component to { r } × M . Furthermore, since the set of critical points ofthe function a ◦ u : S → [0 , r ] is empty, we know that any gradient trajectory in thelower boundary component u − ( { } × M ) will terminate in the upper boundarycomponent u − ( { r } × M ). This is a fact we will heavily exploit.Next, for each x, y ∈ [0 , r ] with x < y we define S yx := { ζ ∈ S : x ≤ a ◦ u ( ζ ) ≤ y } , and α := − ( u ∗ da ) ◦ j = u ∗ λ, as well as the functions h ( s ) := (cid:90) ( a ◦ u ) − ( s ) α and G ( s ) = (cid:90) S s u ∗ ω, and the Riemannian metric γ := u ∗ g. We note that h , G , and γ are smooth.With these definitions in place, we next recall a few linear algebra and calculusfacts. The first is that there exists a large constant C > u for which (cid:107) dα (cid:107) γ ≤ C. This is readily seen here by recalling that α = u ∗ λ , and γ = u ∗ g , so that (cid:107) dα (cid:107) γ = (cid:107) u ∗ dλ (cid:107) u ∗ g ≤ (cid:107) dλ (cid:107) g ≤ C . Next is the fact that given a two-dimensional orientedRiemannian manifold, like ( S, γ ), there exists a corresponding two-dimensionalHausdorff measure dµ γ ; similarly for other dimensions. Furthermore, because ( J, g )is suitably adapted to η , and because u is pseudoholomorphic, we findArea γ ( S ) = (cid:90) S dµ γ = (cid:90) S u ∗ da ∧ α + u ∗ ω, with similar statements for subdomains in S . Finally, the following result is animmediate application of the co-area formula (see Lemma 4.14 below), or a suitably applied change of coordinates. (cid:90) S yx ( u ∗ da ) ∧ α = (cid:90) yx (cid:16) (cid:90) ( a ◦ u ) − ( s ) α (cid:17) ds With these preliminaries established, we now establish our principle differentialinequality. | h (cid:48) ( s ) | = (cid:12)(cid:12)(cid:12) lim (cid:15) → + (cid:15) − (cid:0) h ( s + (cid:15) ) − h ( s ) (cid:1)(cid:12)(cid:12)(cid:12) = lim (cid:15) → + (cid:15) − (cid:12)(cid:12)(cid:12) (cid:90) ( a ◦ u ) − ( s + (cid:15) ) α − (cid:90) ( a ◦ u ) − ( s ) α (cid:12)(cid:12)(cid:12) = lim (cid:15) → + (cid:15) − (cid:12)(cid:12)(cid:12) (cid:90) S s + (cid:15)s dα (cid:12)(cid:12)(cid:12) ≤ lim (cid:15) → + (cid:15) − (cid:90) S s + (cid:15)s (cid:107) dα (cid:107) γ dµ γ ≤ lim (cid:15) → + (cid:15) − C (cid:90) S s + (cid:15)s dµ γ = C lim (cid:15) → + (cid:15) − (cid:90) S s + (cid:15)s (cid:0) ( u ∗ da ) ∧ α + u ∗ ω (cid:1) = C (cid:16) lim (cid:15) → + (cid:15) − (cid:90) S s + (cid:15)s ( u ∗ da ) ∧ α + lim (cid:15) → + (cid:15) − (cid:90) S s + (cid:15)s u ∗ ω (cid:17) = C (cid:0) (cid:90) ( a ◦ u ) − ( s ) α + G (cid:48) ( s ) (cid:1) = C (cid:0) h ( s ) + G (cid:48) ( s ) (cid:1) Or to put it succinctly and in a more useable form, h (cid:48) ( s ) ≤ C (cid:0) h ( s ) + G (cid:48) ( s ) (cid:1) , where C depends on ambient geometry, but not on the map u . Integrating up, wefind h ( s ) ≤ h (0) + C (cid:90) s h ( t ) dt + C (cid:0) G ( s ) − G (0) (cid:1) ≤ (cid:0) h (0) + C (cid:90) S u ∗ ω (cid:1) + C (cid:90) s h ( t ) dt By Gronwall’s inequality (see Lemma 4.15 below), we then have h ( s ) ≤ (cid:0) h (0) + C (cid:90) S u ∗ ω (cid:1) e Cs , or rewriting it, making use of the definition of h , and the fact that α = u ∗ λ , wehave (cid:90) ( a ◦ u ) − ( s ) u ∗ λ ≤ (cid:16) (cid:90) ( a ◦ u ) − (0) u ∗ λ + C (cid:90) S u ∗ ω (cid:17) e Cs . In essence, this is precisely the desired inequality which establishes Theorem 3.To emphasize the key characteristics, the above inequality says that if we considerthe function s (cid:55)→ (cid:82) ( a ◦ u ) − ( s ) u ∗ λ , then the function is bounded from above by s (cid:55)→ Ae Cs , where C depends only on ambient geometry, and A is bounded in terms ofambient geometry constant C , the ω -energy (which is always a priori bounded), ERAL CURVES AND MINIMAL SETS 57 and (cid:82) ( a ◦ u ) − (0) u ∗ λ . Essentially then, we have an exponential bound on the growthof the function s (cid:55)→ (cid:82) ( a ◦ u ) − ( s ) u ∗ λ . To obtain a similar exponential bound on thearea, it is sufficient to recall thatArea u ∗ g ( S r ) = (cid:90) S r u ∗ ( da ∧ λ + ω )= (cid:90) S r u ∗ da ∧ λ + (cid:90) S r u ∗ ω = (cid:90) r (cid:16) (cid:90) ( a ◦ u ) − ( t ) u ∗ λ (cid:1) dt + (cid:90) S r u ∗ ω and then employ our previous exponential growth estimate for s (cid:55)→ (cid:82) ( a ◦ u ) − ( s ) u ∗ λ .We note that while all of the above estimates assume that s ∈ [0 , r ], as similarconstruction and analysis establish the case that s ∈ [ − r, S , the set of regular values of a ◦ u has full measure, and it must be open since the set of critical points is closed. Abit of elementary measure theory then lets us approximate the set of regular valuesfrom the inside by a finite set of compact intervals on which the desired estimateholds. Making use of the fact that ω evaluates non-negatively on J -complex linesand (cid:82) S u ∗ ω < ∞ , and some elementary real analysis then allows us to conclude thedesired inequality for the more general surface. We carry out these details below,but for the moment we sketch further generalizations.Already, such an exponential growth bound on area is rather useful, howeverthere are two more related results which prove to be quite important, and eachessentially stems from the fact that gradient-flow type coordinates are more usefulto us than holomorphic coordinates. More specifically, one can construct localcoordinates ( s, t ) on S , with the property that a ◦ u ( s, t ) = t , and the map t (cid:55)→ ( s , t ) ∈ S is contained in an integral curve of the vector field ∇ ( a ◦ u ). One can thenask if our exponential growth bound on (cid:82) u ∗ λ and the area holds on such rectangularpatches ( s, t ) ∈ [0 , b ] × [0 , r ] of pseudoholomorphic curve. As it turns out, the answeris yes, essentially because 0 = − ( u ∗ da )( j ∇ ( a ◦ u )) = α ( ∇ ( a ◦ u )) = u ∗ λ ( ∇ ( a ◦ u )).Indeed, replacing S yx with (cid:101) S yx := { ζ ∈ S yx : 0 ≤ s ( ζ ) ≤ b } where ( s, t ) are rectangular gradient-like coordinates as above, we see the entireargument carries over unchanged, including (cid:90) ( a ◦ u ) − ( s + (cid:15) ) α − (cid:90) ( a ◦ u ) − ( s + (cid:15) ) α = (cid:90) (cid:101) S s + (cid:15)s dα. This latter equality holds precisely because α ( ∇ ( a ◦ u )) = 0, and this guaranteesthat there are no contributions to Stokes’ theorem coming from the gradient-like“sides” of our pseudoholomorphic rectangle. Essentially then, this establishes ex-ponential area growth for certain pseudoholomorphic rectangles, which we definemore precisely as tracts of pseudoholomorphic curves in Definition 4.2 below.The final generalization of our exponential growth bound is less enlighteningand more a necessary evil. The issue is that in later sections, we will need to studyportions of a pseudoholomorphic curve restricted to S x + (cid:15)x where x and x + (cid:15) are regular values of a ◦ u . Moreover, we would like to claim that if (cid:82) ( a ◦ u ) − ( x ) u ∗ λ isvery large, and if (cid:82) S x + (cid:15)x u ∗ ω is very small, then most of the gradient trajectorieswhich start along the set ( a ◦ u ) − ( x ) terminate at a point in ( a ◦ u ) − ( x + (cid:15) ). As itturns out, this is not difficult to establish in the special case that the function a ◦ u is Morse, but it appears to be intractable in the general case. This forces us intothe position that we must establish the desired exponential growth bound on areafor perturbed pseudoholomorphic curves – that is, for curves which are no longerpseudoholomorphic. Worse still, the type of perturbation, and specifically its precise size , will be important for later estimates, so we must establish the desired areaestimate for all perturbed pseudoholomorphic curves for which the perturbationis small in a very explicit manner. In turn, this seems to force us to establish anumber of rather elementary estimates via rather tedious but elementary means,and this takes up a bulk of the Section 4.1. The upshot however, is that we establishthe desired estimates for perturbed curves, which is crucial for later results. Theremainder of Section 4.1 is then devoted to making these above sketches rigorous.4.1.2. Definitions and Elementary Estimates.
Definition 4.1 (perturbed pseudoholomorphic map) . Let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) , and let ( J, g ) be an η -adapted almost Hermitian structure on R × M . A perturbed pseudoholomorphicmap consists of the tuple (˜ u, ˜ , f, u, S, j ) where(p1) u : ( S, j ) → ( R × M, J ) is a generally immersed pseudoholomorphic map,which is possibly non-compact,(p2) f : S → R is a smooth function,(p3) the support of f is compact and satisfies supp( f ) ⊂ S \ ( ∂S ∪ Z ) , where Z = { ζ ∈ S : T u ( ζ ) = 0 } (p4) ˜ u ( ζ ) = exp gu ( ζ ) ( f ( ζ ) ∂ a ) , where exp g is the exponential map associated to theRiemannian metric g , and ∂ a is the coordinate vector field associated to thecoordinate a ∈ R ,(p5) ˜ is a smooth almost complex structure on S which induces the same orienta-tion as j ,(p6) on the complement of supp( f ) we have j = ˜ , and elsewhere ˜ is uniquelydetermined by requiring that ˜ is a ˜ u ∗ g -isometry. Geometrically then, a perturbed pseudoholomorphic map is obtained by nudgingan honestly pseudoholomorphic map a bit in the symplectization direction. Werequire this modification to be away from the boundary of S and critical pointsof u , and in practice it will occur only in a small neighborhood of the criticalpoints of a ◦ u . We adapt the almost complex structure on the domain so thatour new perturbed map is an isometry, but not pseudoholomorphic. For notationalconvenience and ease of exposition, rather than write the full tuple (˜ u, ˜ , f, u, S, j )to specify a perturbed pseudoholomorphic map, we will instead say: Let (˜ u, S, ˜ )be an f -perturbation of a pseudoholomorphic map ( u, S, j ). Definition 4.2 (tract of perturbed pseudoholomorphic map) . Let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) , and let ( J, g ) be an η -adapted almost Hermitian structure on R × M . A tract of perturbed pseudoholo-morphic map consists of the tuple (˜ u, (cid:101) S, ˜ , f, u, S, j ) where ERAL CURVES AND MINIMAL SETS 59 (1) (˜ u, ˜ , f, u, S, j ) is a perturbed pseudoholomorphic map,(2) (cid:101) S ⊂ S is a smooth real two dimensional non-empty manifold, possibly withboundary, possibly with corners, and possibly non-compact,(3) the restriction ˜ u : (cid:101) S → R × M is a proper map satisfying { ζ ∈ S : d ( a ◦ ˜ u ) ζ = 0 } ∩ ∂ (cid:101) S = ∅ , (4) the boundary of (cid:101) S decomposes as ∂ (cid:101) S = ∂ (cid:101) S ∪ ∂ (cid:101) S where(a) the set ∂ (cid:101) S ∩ ∂ (cid:101) S is finite(b) along ∂ (cid:101) S the vector field (cid:101) ∇ ( a ◦ ˜ u ) is tangent to ∂ (cid:101) S ; here (cid:101) ∇ is thegradient computed with respect to the metric ˜ γ := ˜ u ∗ g ,(c) the restriction of the map a ◦ ˜ u to each connected component of ∂ (cid:101) S isa constant map. Geometrically, a tract of perturbed pseudoholomorphic map is a perturbed pseu-doholomorphic map with boundary and corners, with the property that the bound-ary is piecewise smooth, and each smooth portion is either a level set of ( a ◦ u orelse an integral curve of (cid:101) ∇ a ◦ ˜ u . We denote the level-set type boundary by ∂ (cid:101) S , andwe denote the gradient-line type boundary by ∂ (cid:101) S . The corners of the boundaryare those points where the two types of boundary intersect. Definition 4.3 (the characteristic α -foliation: F α ) . Let S be a real -dimensional manifold, possibly with boundary, possibly with cor-ners, and possibly non-compact. Suppose α ∈ Ω ( S ) is a smooth one-form on S .Then we define the characteristic α -foliation, F α ⊂ T S , by F α = (cid:91) ζ ∈ S F αζ where F αζ ⊂ T ζ S is given by F αζ = (cid:40) ker α ζ if dim(ker α ζ ) = 10 otherwise . Lemma 4.4 (characteristic ˜ α -foliation is gradient) . Let ( M, η = ( λ, ω )) be a framed Hamiltonian manifold, let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M , let (˜ u, S, ˜ ) be an f -perturbation of a pseu-doholomorphic map ( u, S, j ) Definition 4.1, and let ˜ α = − d ( a ◦ ˜ u ) ◦ ˜ be as above.Then (13) { ζ ∈ (cid:101) S : dim ( F ˜ αζ ) = 0 } = { ζ ∈ (cid:101) S : d ( a ◦ u ) ζ = 0 } , and (cid:101) ∇ ( a ◦ ˜ u ) , when thought of as a subset of T (cid:101) S and computed with respect to themetric ˜ γ = ˜ u ∗ g satisfies the property (14) (cid:101) ∇ ( a ◦ ˜ u ) ⊂ F ˜ α . Proof.
We first observe that ˜ α is a smooth one-form on a two-manifold, and hencedim(ker ˜ α ζ ) ∈ { , } . The set of points where this dimension is two, is preciselythe set of points where ˜ α is zero – however, 0 = ˜ α ζ = d ( a ◦ ˜ u ) ζ ◦ ˜ , and since ˜ is a˜ γ -isometry, we see that the set where this dimension is two is precisely the set ofcritical points of a ◦ ˜ u . Equation (13) follows immediately.To establish (14), we observe that˜ α (cid:0) (cid:101) ∇ ( a ◦ ˜ u ) (cid:1) = − d ( a ◦ ˜ u ) (cid:0) ˜ (cid:101) ∇ ( a ◦ ˜ u ) (cid:1) = 0 By proper, we mean that for each compact set
K ⊂ R × M the set ˜ u − ( K ) is compact. where to obtain the second equality we have used the fact that the almost complexstructure ˜ is a ˜ γ -isometry and (cid:101) ∇ is the gradient with respect to ˜ γ . (cid:3) Given an f -perturbation of a pseudoholomorphic map, it will be important tohave certain properties of the metric ˜ γ = ˜ u ∗ g estimated in terms of properties ofthe metric γ = u ∗ g . As such, we have the following. Lemma 4.5 (˜ γ -estimates) . Let < (cid:15) < − , let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) ,and let ( J, g ) be an η -adapted almost Hermitian structure on W := R × M , andlet ( u, S, j ) be a generally immersed pseudoholomorphic map. Let f be a smoothfunction, and let ˜ u = (˜ u, S, ˜ ) be an f -perturbation of ( u, S, j ) . Suppose further that (cid:107) df (cid:107) γ + (cid:107)∇ df (cid:107) γ ≤ (cid:15) (1 + (cid:107) B u (cid:107) γ ) here ∇ denotes covariant differentiation with respect to the Levi-Civita connectionassociated to the metric γ = u ∗ g , B u denotes the second fundamental form of u : S → R × M as given in Definition A.4, and finally by (cid:107) df (cid:107) γ , (cid:107)∇ df (cid:107) γ , and (cid:107) B u (cid:107) γ we respectively mean the L ∞ norm of each over the support of f . Recall that Z is the set of singular points of u , i.e. Z = { ζ ∈ S | T u ( ζ ) = 0 } . Then for anyvector fields
Y, Z ∈ Γ( T S → ( S \ Z )) we have (15) (cid:12)(cid:12) (cid:104) Y, Z (cid:105) γ − (cid:104) Y, Z (cid:105) ˜ γ (cid:12)(cid:12) ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ and (16) (cid:107)∇ Y Z − (cid:101) ∇ Y Z (cid:107) γ ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ . And for each one-form α on S we have (17) (cid:107)∇ Y α − (cid:101) ∇ Y α (cid:107) γ ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) α (cid:107) γ . Proof.
Since the inequalities are trivially true for any ζ ∈ S \ supp( f ), we be-gin by fixing a point ζ ∈ supp( f ), and letting ( y , y ) denote γ -normal geodesiccoordinates centered at ζ . Next, we let ( x , . . . , x m − ) denote normal geodesiccoordinates on M associated to metric λ ⊗ λ + ω ◦ ( × Jπ ξ ) and centered at thepoint u ( ζ ); here π ξ is the projection along the line bundle ker ω → M to the hy-perplane distribution ξ := ker λ . Define the coordinate x := a where a is the usualsymplectization coordinate on R . We simplify the following computation using theabbreviation ˜ u i = x i ◦ ˜ u and ˜ u ,i = T u · ∂ y i , and then˜ γ k(cid:96) = (cid:104) ˜ u ,k , ˜ u ,(cid:96) (cid:105) g = g ij ˜ u i,k ˜ u j,(cid:96) = g ij ( u i,k + δ i f ,k )( u j,(cid:96) + δ j f ,(cid:96) )= g ij u i,k u j,(cid:96) + g ij δ i f ,k u j,(cid:96) + g ij u i,k δ j f ,(cid:96) + g ij δ i f ,k δ j f ,(cid:96) = g ij u i,k u j,(cid:96) + g j f ,k u j,(cid:96) + g i u i,k f ,(cid:96) + g f ,k f ,(cid:96) = g ij u i,k u j,(cid:96) + f ,k u ,(cid:96) + u ,k f ,(cid:96) + f ,k f ,(cid:96) = γ k(cid:96) + f ,k u ,(cid:96) + u ,k f ,(cid:96) + f ,k f ,(cid:96) ;where throughout the above computation g ij is evaluated at ˜ u ( y , y ), and g ij u i,k u j,(cid:96) = γ k(cid:96) because the g ij are independent of the x coordinate; that is, the metric g is in-dependent of translation in the symplectization direction. To be clear, by definition ERAL CURVES AND MINIMAL SETS 61 it is true that g ij ( u ( y , y )) u i,k ( y , y ) u j,l ( y , y ) = γ kl ( y , y ) , however we are claiming g ij (˜ u ( y , y )) u i,k ( y , y ) u j,l ( y , y ) = γ kl ( y , y ) , which is only true because ˜ u ( y , y ) = u ( y , y )+ (cid:0) f ( y , y ) , , . . . , (cid:1) and g is invari-ant under R -shifts. Furthermore, since ( y , y ) are γ -normal geodesic coordinates,it follows that γ ij,k ( ζ ) = 0. Consequently(18) ˜ γ k(cid:96),n ( ζ ) = ( f ,kn u ,(cid:96) + f ,k u ,(cid:96)n + u ,kn f ,(cid:96) + u ,k f ,(cid:96)n + f ,kn f ,(cid:96) + f ,k f ,(cid:96)n ) (cid:12)(cid:12) ζ It will be convenient to evaluate a few functions at ζ ; making use of the fact that( y , y ) is a γ -normal geodesic coordinate system, together with Lemma A.1 andthe fact that (cid:107) dx (cid:107) g = 1 the following inequalities are straightforward to verify. | f ,k ( ζ ) | ≤ (cid:107) df (cid:107) γ (19) | f ,k(cid:96) ( ζ ) | ≤ (cid:107)∇ df (cid:107) γ (20) | u ,k ( ζ ) | ≤ . (21)Now we make use of some elementary results from Riemannian geometry whichare elaborated upon in Appendix A.1, specifically Lemma A.1, Corollary A.2, andDefinition A.4, and we establish the following. | u ,k(cid:96) ( ζ ) | = (cid:12)(cid:12) ∂ y (cid:96) (cid:0) dx ( u ,k ) (cid:1)(cid:12)(cid:12) ζ (cid:12)(cid:12) ≤ | ( ∇ u ,(cid:96) dx )( u ,k ) | + | dx ( ∇ u ,(cid:96) u ,k ) |≤ | ( ∇ u ,(cid:96) dx )( u ,k ) | + (cid:107)∇ u ,(cid:96) u ,k (cid:107) g ≤ (cid:107) ( ∇ u ,(cid:96) u ,k ) (cid:62) (cid:107) g + (cid:107) ( ∇ u ,(cid:96) u ,k ) ⊥ (cid:107) g = (cid:107)∇ ∂ y(cid:96) ∂ y k (cid:107) u ∗ g + (cid:107) B u ( u ,k , u ,(cid:96) ) (cid:107) g ≤ (cid:107) B u (cid:107) g . (22)With this inequality established, we let Y = Y i ∂ y i and Z = Z i ∂ y i , and establishinequality (15). The following functions, forms, vector fields, etc, are all assumedto be evaluated at ζ . (cid:12)(cid:12) (cid:104) Y, Z (cid:105) ˜ γ − (cid:104) Y, Z (cid:105) γ (cid:12)(cid:12) = (cid:12)(cid:12) ˜ γ k(cid:96) Y k Z (cid:96) − γ k(cid:96) Y k Z (cid:96) (cid:12)(cid:12) ≤ (cid:0) | f ,k u ,(cid:96) | + | u ,k f ,(cid:96) | + | f ,k f ,(cid:96) | (cid:1) | Y k || Z (cid:96) |≤ (cid:0) (cid:107) df (cid:107) γ + (cid:107) df (cid:107) γ (cid:1) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ ;this establishes inequality (15). To establish inequality (16), we make use of theformula for the Christoffel symbols given in equation (131) as well as the factthat the Christoffel symbols vanish at the center of normal geodesic coordinates toestimate the following; again all functions, forms, etc are assumed to be evaluatedat ζ . (cid:107)∇ Y Z − (cid:101) ∇ Y Z (cid:107) γ = (cid:107) Y i Z j (cid:101) Γ kij ∂ y k (cid:107) γ ≤ (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ max i,j,k | (cid:101) Γ kij | See for instance equation (134). = 2 (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ max i,j,k (cid:12)(cid:12) ˜ γ k(cid:96) (cid:0) ˜ γ i(cid:96),j + ˜ γ j(cid:96),i − ˜ γ ij,(cid:96) (cid:1)(cid:12)(cid:12) ≤ (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ (cid:0) max i,j | ˜ γ ij | (cid:1)(cid:0) max i,j,k | ˜ γ ij,k | (cid:1) . (23)To continue, we make use of inequality (15), which guarantees that | δ ij − ˜ γ ij | ≤ (cid:15) ,and estimate max i,j | ˜ γ ij | ≤ max i,j | ˜ γ ij | ˜ γ ˜ γ − ˜ γ ˜ γ ≤ (cid:15) (1 − (cid:15) ) − (cid:15) ≤ . (24)By combining inequalities (18) - (22) we have | ˜ γ ij,k ( ζ ) | ≤ (cid:107)∇ df (cid:107) γ + 2 (cid:107) df (cid:107) γ (cid:107) B u (cid:107) γ + 2 (cid:107) df (cid:107) γ (cid:107)∇ df (cid:107) γ ≤ (cid:16) (cid:15) ( (cid:107) B u (cid:107) γ + 1) + (cid:15) (cid:107) B u (cid:107) γ ( (cid:107) B u (cid:107) γ + 1) + (cid:15) ( (cid:107) B u (cid:107) γ + 1) (cid:17) ≤ (cid:15) . Combining this with inequalities (23) and (24) then yields (cid:107)∇ Y Z − (cid:101) ∇ Y Z (cid:107) γ ≤ · (cid:15) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) Z (cid:107) γ . This establishes inequality (16). Finally, recall (e.g. from Section A.1) that (cid:101) ∇ ∂ yk dy i = − (cid:101) Γ (cid:96)ki dy (cid:96) and at ζ we have ∇ ∂ yk dy i = 0. Consequently, if α ∈ Ω ( S ) is a one-formwritten in coordinates as α = α (cid:96) dy (cid:96) , then at ζ we have (cid:107)∇ Y α − (cid:101) ∇ Y α (cid:107) γ = (cid:107) Y i α j (cid:101) Γ kij dx k (cid:107) γ ≤ (cid:15) (cid:107) Y (cid:107) γ (cid:107) α (cid:107) γ as above. This establishes inequality (17), and completes the proof of Lemma4.5. (cid:3) Given a framed Hamiltonian manifold (
M, η ) and an f -perturbation (˜ u, S, ˜ ) ofa generally immersed pseudoholomorphic map ( u, S, j ) in R × M with ( J, g ) an η -adapted almost Hermitian structure, we define a one-form ˜ α by the following.(25) ˜ α ∈ Ω ( S ) by ˜ α := − (˜ u ∗ da ) ◦ ˜ Lemma 4.6 ( d ˜ α estimate) . Let ˜ α be the one-form defined by equation (25). Then (26) d ˜ α = (cid:0) (cid:101) ∆( a ◦ ˜ u ) (cid:1) vol ˜ γ where (cid:101) ∆ is the Laplace-Beltrami operator given by (cid:101) ∆ f = tr( (cid:101) ∇ df )= ˜ γ ij (cid:16) (cid:101) ∇ ∂ xi ( (cid:101) ∇ ∂ xj f ) − (cid:101) ∇ (cid:101) ∇ ∂xi ∂ xj f (cid:17) and vol ˜ γ is the volume form associated to ˜ γ and the orientation on S . Conse-quently, the following pointwise equality holds (27) (cid:107) d ˜ α (cid:107) ˜ γ = | (cid:101) ∆( a ◦ ˜ u ) | . See Section A.1 for more details, specifically equation (137).
ERAL CURVES AND MINIMAL SETS 63
Proof.
Fix ζ ∈ S , and let (˜ s, ˜ t ) denote ˜ -holomorphic coordinates centered at ζ for which (cid:107) ∂ ˜ s (cid:107) ˜ γ = 1 = (cid:107) ∂ ˜ t (cid:107) ˜ γ at ζ . Then in these coordinates we have˜ α = ˜ α ( ∂ ˜ s ) d ˜ s + ˜ α ( ∂ ˜ t ) d ˜ t = − da ( T ˜ u · ˜ · ∂ ˜ s ) d ˜ s − da ( T ˜ u · ˜ · ∂ ˜ t ) d ˜ t = − da ( T ˜ u · ∂ ˜ t ) d ˜ s + da ( T ˜ u · ∂ ˜ s ) d ˜ t = − ( a ◦ ˜ u ) ˜ t d ˜ s + ( a ◦ ˜ u ) ˜ s d ˜ t. Consequently d ˜ α = (cid:0) ( a ◦ ˜ u ) ˜ s ˜ s + ( a ◦ ˜ u ) ˜ t ˜ t (cid:1) d ˜ s ∧ d ˜ t. Next we note ( a ◦ ˜ u ) ˜ s ˜ s = (cid:101) ∇ ∂ ˜ s ( a ◦ ˜ u ) ˜ s = (cid:101) ∇ ∂ ˜ s (cid:0) d ( a ◦ ˜ u )( ∂ ˜ s ) (cid:1) = (cid:0) (cid:101) ∇ ∂ ˜ s d ( a ◦ u ) (cid:1) ( ∂ ˜ s ) + d ( a ◦ ˜ u ) (cid:0) (cid:101) ∇ ∂ ˜ s ∂ ˜ s (cid:1) . and thus d ˜ α = (cid:16)(cid:0) (cid:101) ∇ ∂ ˜ s d ( a ◦ u ) (cid:1) ( ∂ ˜ s ) + (cid:0) (cid:101) ∇ ∂ ˜ t d ( a ◦ u ) (cid:1) ( ∂ ˜ t ) (cid:17) d ˜ s ∧ d ˜ t + (cid:16) d ( a ◦ ˜ u ) (cid:0) (cid:101) ∇ ∂ ˜ s ∂ ˜ s + (cid:101) ∇ ∂ ˜ t ∂ ˜ t (cid:1)(cid:17) d ˜ s ∧ d ˜ t = (cid:16)(cid:0) (cid:101) ∇ ∂ ˜ s d ( a ◦ ˜ u ) (cid:1) ( ∂ ˜ s ) + (cid:0) (cid:101) ∇ ∂ ˜ t d ( a ◦ ˜ u ) (cid:1) ( ∂ ˜ t ) (cid:17) d ˜ s ∧ d ˜ t where the final equality follows from Lemma 4.7 below. Finally, we evaluate thisequality at ζ and make use of the fact that at ζ the ordered pair ( ∂ ˜ s , ∂ ˜ t ) is apositively oriented ˜ γ -orthonormal basis to conclude that indeed d ˜ α = (cid:0) (cid:101) ∆( a ◦ ˜ u ) (cid:1) vol ˜ γ . This establishes equation (26); equation (27) follows immediately. (cid:3)
Lemma 4.7 (conformal coordinates property) . Let S be a smooth real -dimensional manifold, equipped with an almost Hermitianstructure ( j, γ ) . Suppose further that ∇ denotes covariant differentiation with re-spect to the Levi-Civita connection associated to the metric γ , and ( s, t ) are localcoordinates for which j∂ s = ∂ t . Then ∇ ∂ s ∂ s + ∇ ∂ t ∂ t = 0 . Proof.
Let ( x , x ) = ( s, t ), so that γ = γ ij dx i ⊗ dx j . Observe that because ( j, γ ) isalmost Hermitian, we have (cid:104) ∂ s , ∂ t (cid:105) γ = (cid:104) ∂ s , j∂ s (cid:105) γ = 0 and (cid:104) ∂ s , ∂ s (cid:105) γ = (cid:104) j∂ s , j∂ s (cid:105) γ = (cid:104) ∂ t , ∂ t (cid:105) γ . Consequently γ = h ds + h dt , where h is a smooth positive function depending on s and t . Let Γ kij be denote theChristoffel symbols associated to the Levi-Civita connection corresponding to thismetric. Recall these are given byΓ kij = 12 γ k(cid:96) (cid:0) γ i(cid:96),j + γ j(cid:96),i − γ ij,(cid:96) ) where γ ij,k = ∂∂x k γ ij . We compute ∇ ∂ s ∂ s + ∇ ∂ t ∂ t = (cid:0) Γ ∂ s + Γ ∂ t (cid:1) + (cid:0) Γ ∂ s + Γ ∂ t (cid:1) = (Γ + Γ ) ∂ s + (Γ + Γ ) ∂ t ; however, Γ = 12 γ ( γ , + γ , − γ , ) = 12 h ∂∂x h Γ = 12 γ ( γ , + γ , − γ , ) = − h ∂∂x h Γ = 12 γ ( γ , + γ , − γ , ) = − h ∂∂x h Γ = 12 γ ( γ , + γ , − γ , ) = 12 h ∂∂x h The desired result is immediate. (cid:3)
Lemma 4.8 (coercive estimate) . Let < (cid:15) < − , let ( M, η = ( λ, ω )) be a framed Hamiltonian manifold, let ( J, g ) be an η -adapted almost Hermitian structure on W := R × M , and let u : ( S, j ) → ( W, J ) be a J -holomorphic map which is also an immersion. Suppose further that ˜ u = (˜ u, S, ˜ ) is an f -perturbation of ( u, S, j ) , where f : S → R is a smooth functionsatisfying (cid:107) df (cid:107) γ + (cid:107)∇ df (cid:107) γ ≤ (cid:15) (1 + (cid:107) B u (cid:107) γ ) ; here ∇ denotes covariant differentiation with respect to the Levi-Civita connectionassociated to the metric γ = u ∗ g , B u denotes the second fundamental form of u : S → R × M as given in Definition A.4, and finally by (cid:107) df (cid:107) γ , (cid:107)∇ df (cid:107) γ , and (cid:107) B u (cid:107) γ we respectively mean the L ∞ norm of each. Then , for any ˜ γ -unit vector τ ∈ T S we have ≤ (cid:0) (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω (cid:1) ( τ, ˜ τ ) . Proof.
To begin, we recall that as a consequence of Lemma 4.5, we have(1 − (cid:15) ) (cid:107) τ (cid:107) ˜ γ ≤ (cid:15) ) (cid:107) τ (cid:107) ˜ γ ≤ (cid:107) τ (cid:107) γ ≤ − (cid:15) ) (cid:107) τ (cid:107) ˜ γ ≤ (1 + 2 (cid:15) ) (cid:107) τ (cid:107) ˜ γ for any τ ∈ T S ; here we have made use of the elementary inequalities (1 − (cid:15) ) − ≤ (cid:15) and 1 − (cid:15) ≤ (1 + (cid:15) ) − which hold for 0 ≤ (cid:15) ≤ . We henceforth assume (cid:107) τ (cid:107) ˜ γ = 1. Our first task is to estimate (cid:107) j − ˜ (cid:107) γ . To that end, we recall that j and˜ are the almost complex structures uniquely determined by the respective metrics γ and ˜ γ and the orientation on S . Consequently, we can write jτ = (cid:107) τ (cid:107) γ ˜ τ − (cid:104) ˜ τ,τ (cid:105) γ (cid:107) τ (cid:107) γ τ (cid:107) ˜ τ − (cid:104) ˜ τ,τ (cid:105) γ (cid:107) τ (cid:107) γ τ (cid:107) γ . We can then write (cid:107) ( j − ˜ ) τ (cid:107) γ = (cid:107) ( K − τ − KLτ (cid:107) γ . (28)where L = (cid:104) ˜ τ, τ (cid:105) γ (cid:107) τ (cid:107) γ and K = (cid:107) τ (cid:107) γ (cid:107) ˜ τ − Lτ (cid:107) γ . We now estimate the relevant terms. | L | = |(cid:104) ˜ τ, τ (cid:105) γ |(cid:107) τ (cid:107) γ ≤ (cid:15) (cid:107) ˜ τ (cid:107) γ (cid:107) τ (cid:107) γ ≤ (cid:15) (cid:15) − (cid:15) ≤ (cid:15). ERAL CURVES AND MINIMAL SETS 65 | K | ≤ (cid:107) τ (cid:107) γ (cid:107) ˜ τ (cid:107) γ − | L |(cid:107) τ (cid:107) γ ≤ (cid:15) (1 − (cid:15) ) − (cid:15) (1 + 2 (cid:15) ) ≤ . (cid:12)(cid:12) | K | − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:107) τ (cid:107) γ − (cid:107) ˜ τ − Lτ (cid:107) γ (cid:107) ˜ τ − Lτ (cid:107) γ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) (cid:107) τ (cid:107) γ − (cid:12)(cid:12) + (cid:12)(cid:12) (cid:107) ˜ τ − Lτ (cid:107) γ − (cid:12)(cid:12) (1 − (cid:15) ) − (cid:15) (1 + 2 (cid:15) ) ≤ (cid:15) + (1 + 2 (cid:15) ) − (1 − (cid:15) − (cid:15) (1 + 2 (cid:15) ))(1 − (cid:15) ) − (cid:15) (1 + 2 (cid:15) ) ≤ (cid:15). Combining these estimates with equation (28) immediately yields (cid:107) ( j − ˜ ) τ (cid:107) γ ≤ (cid:0) | K − | + | K | | L | (cid:1) (cid:107) τ (cid:107) γ ≤ (cid:15) (1 + 2 (cid:15) ) , and hence (cid:107) j − ˜ (cid:107) γ ≤ (cid:15) (1 + 2 (cid:15) ) ≤ (cid:15). Now making use of the fact that ω ( ∂ a , · ) = 0, and the fact that˜ u ( ζ ) = exp gu ( ζ ) ( f ( ζ ) ∂ a )we find | ˜ u ∗ ω ( τ, ˜ τ ) − u ∗ ω ( τ, jτ ) | = | u ∗ ω ( τ, (˜ − j ) τ ) |≤ (cid:107) u ∗ ω (cid:107) γ (cid:107) τ (cid:107) γ (cid:107) j − ˜ (cid:107) γ ≤ (cid:15) (1 + 2 (cid:15) ) (cid:107) ω (cid:107) g ≤ (cid:15) ≤
110 ;here we have made use of the easily verified estimate (cid:107) ω (cid:107) g ≤
1. To obtain a similarestimate for the ˜ u ∗ da ∧ ˜ α term, we first observe that(˜ u ∗ da ) ∧ (˜ u ∗ da ◦ ˜ )( τ, ˜ τ )= ( df + u ∗ da ) ∧ (cid:0) df ◦ ˜ + u ∗ da ◦ j + u ∗ da ◦ (˜ − j ) (cid:1) ( τ, (˜ − j ) τ )+ ( df + u ∗ da ) ∧ (cid:0) df ◦ ˜ + u ∗ da ◦ j + u ∗ da ◦ (˜ − j ) (cid:1) ( τ, jτ )From this it follows that (cid:12)(cid:12) (˜ u ∗ da ) ∧ (˜ u ∗ da ◦ ˜ )( τ, ˜ τ ) − ( u ∗ da ) ∧ ( u ∗ da ◦ j )( τ, jτ ) |≤ (cid:0) (cid:107) df (cid:107) γ + (cid:107) u ∗ da (cid:107) γ (cid:1)(cid:0) (cid:107) df (cid:107) γ (cid:107) ˜ (cid:107) γ + (cid:107) u ∗ da (cid:107) γ + (cid:107) u ∗ da (cid:107) γ (cid:107) ˜ − j (cid:107) γ (cid:1) (cid:107) ˜ − j (cid:107) γ (cid:107) τ (cid:107) γ + (cid:107) df (cid:107) γ (cid:0) (cid:107) df (cid:107) γ (cid:107) ˜ (cid:107) γ + (cid:107) u ∗ da (cid:107) γ (cid:107) ˜ (cid:107) γ (cid:1) (cid:107) τ (cid:107) γ + (cid:107) u ∗ da (cid:107) γ (cid:0) (cid:107) df (cid:107) γ (cid:107) ˜ (cid:107) γ + (cid:107) u ∗ da (cid:107) γ (cid:107) ˜ − j (cid:107) γ (cid:1) (cid:107) τ (cid:107) γ ≤ (cid:0) (cid:15) + 1 (cid:1)(cid:0) (cid:15) (1 + 100 (cid:15) ) + 1 + 100 (cid:15) (cid:1) (cid:15) (1 + 2 (cid:15) ) + (cid:15) (cid:0) (cid:15) (1 + 100 (cid:15) ) + (1 + 100 (cid:15) ) (cid:1) (1 + 2 (cid:15) ) + (cid:0) (cid:15) (1 + 100 (cid:15) ) + 100 (cid:15) (cid:1) (1 + 2 (cid:15) ) ≤ (cid:15) ≤ . Combining this with our previous estimate for | ˜ u ∗ ω ( τ, ˜ τ ) − u ∗ ω ( τ, jτ ) | then yields (cid:12)(cid:12)(cid:0) (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω (cid:1) ( τ, ˜ τ ) − (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω (cid:1) ( τ, ˜ τ ) − (cid:0) ( u ∗ da ) ∧ α + u ∗ ω (cid:1) ( τ, jτ ) (cid:12)(cid:12) ≤ . where we have written˜ α = − ˜ u ∗ da ◦ ˜ and α = − u ∗ da ◦ j = u ∗ λ. This is the desired estimate, and hence completes the proof of Lemma 4.8. (cid:3)
Before proceeding, we must make some quick estimates of dλ and establish somegeometric constants. This is the purpose of Lemma 4.9 and Definition 4.11, below. Lemma 4.9 ( dλ bounds) . Let ( M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ) , let ( J, g ) be an η -adapted almost Hermitian structure on R × M , and fix Y ∈ T ( R × M ) . Then | dλ ( Y, JY ) | ≤ C ( da ∧ λ + ω )( Y, JY ) where C = (cid:107) i X η dλ (cid:12)(cid:12) ξ (cid:107) g + (cid:107) dλ (cid:12)(cid:12) ξ (cid:107) g (29) where (cid:107) i X η dλ (cid:12)(cid:12) ξ (cid:107) g = sup Y ∈ ξY (cid:54) =0 | dλ ( X η , Y ) | ( ω ( Y, JY )) and (cid:107) dλ (cid:12)(cid:12) ξ (cid:107) g = sup Y ∈ ξY (cid:54) =0 | dλ ( Y, JY ) | ω ( Y, JY ) . Proof.
First, write Y = Y + Y with Y ∈ Span( ∂ a , X η ) and Y ∈ ξ . Then | dλ ( Y, JY ) | ≤ | dλ ( Y , JY ) | + | dλ ( Y , JY ) | + | dλ ( Y , JY ) | + | dλ ( Y , JY ) | = | dλ ( Y , JY ) | + | dλ ( Y , JY ) | + | dλ ( Y , JY ) | = | dλ ( λ ( Y ) X η , JY ) | + | dλ ( Y , λ ( JY ) X η ) | + | dλ ( Y , JY ) | = | λ ( Y ) || dλ ( X η , JY ) | + | da ( Y ) || dλ ( Y , X η ) | + | dλ ( Y , JY ) |≤ | λ ( Y ) | c (cid:0) ω ( Y , JY ) (cid:1) + | da ( Y ) | c (cid:0) ω ( Y , JY ) (cid:1) + c ω ( Y , JY )= | λ ( Y ) | c (cid:0) ω ( Y, JY ) (cid:1) + | da ( Y ) | c (cid:0) ω ( Y, JY ) (cid:1) + c ω ( Y, JY ) ≤ c (cid:0) | λ ( Y ) | + | da ( Y ) | (cid:1) + ( c + c ) ω ( Y, JY )= c ( da ∧ λ )( Y, JY ) + ( c + c ) ω ( Y, JY ) ≤ C ( da ∧ λ + ω )( Y, JY ) , where to obtain the final equality we have made use of Lemma 2.6. This is thedesired estimate. (cid:3) ERAL CURVES AND MINIMAL SETS 67
Remark 4.10 (deviation from stable) . We note that the proof of Lemma 4.9 immediately establishes the more preciseestimate | dλ ( Y, JY ) | ≤ c ( da ∧ λ )( Y, JY ) + ( c + c ) ω ( Y, JY ) where c = (cid:107) i X η dλ (cid:12)(cid:12) ξ (cid:107) g and c = (cid:107) dλ (cid:12)(cid:12) ξ (cid:107) g . We do not use this additional precision inthis manuscript, however it highlights the fact that c becomes a means to measurethe degree to which ( λ, ω ) fails to be a stable Hamiltonian structure. That is, c = 0 if and only if ( λ, ω ) is a stable Hamiltonian structure, and in some sense, the larger c is the further our Hamiltonian structure deviates from being stable. Definition 4.11 (ambient geometry constant) . For each tuple h = ( M, λ, ω, J, g ) , where M is a closed odd dimensional manifold, η = ( λ, ω ) is a framed Hamiltonian structure on M , and ( J, g ) is an η -adaptedalmost Hermitian structure on R × M , we define the following finite number: C h := 2 (cid:0)
10 + max(1 , c + c ) (cid:1) where c = (cid:107) i X η dλ (cid:12)(cid:12) ξ (cid:107) g and c = (cid:107) dλ (cid:12)(cid:12) ξ (cid:107) g as above in Lemma 4.9. We now proceed with our final elementary estimate.
Lemma 4.12 ( d ˜ α bounds) . Let ( M, η = ( λ, ω )) be a framed Hamiltonian manifold, let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M , and let (˜ u, ˜ , f, u, S, j ) be a perturbed pseu-doholomorphic map. Let < (cid:15) < min(2 − , (1 + sup ζ ∈ supp( f ) (cid:107) B u ( ζ ) (cid:107) γ ) − ) . Suppose further that (cid:107) df (cid:107) γ + (cid:107)∇ df (cid:107) γ ≤ (cid:15) (1 + (cid:107) B u (cid:107) γ ) , where (cid:107) df (cid:107) γ , (cid:107)∇ df (cid:107) γ , and (cid:107) B u (cid:107) γ are the L ∞ norms over the support of f . Thensup ζ ∈ S (cid:107) d ˜ α ζ (cid:107) ˜ γ ≤ C h where C h is the ambient geometry constant associated to h = ( M, λ, ω, J, g ) and asprovided in Definition 4.11.Proof. First we note that for ζ / ∈ supp( f ), we have ˜ u = u and ˜ α = u ∗ λ , and then itfollows from Lemma 4.9 that (cid:107) d ˜ α ζ (cid:107) γ = (cid:107) u ∗ dλ ζ (cid:107) γ ≤ C h . As such we will assumefor the remainder of the proof that ζ ∈ supp( f ), and for notational clarity weremove ζ from the notation.In light of Lemma 4.6, we have (cid:107) d ˜ α (cid:107) ˜ γ = | (cid:101) ∆( a ◦ ˜ u ) | and (cid:107) dα (cid:107) γ = | ∆( a ◦ u ) | . In order to estimate further, fix ζ ∈ S , and choose γ -orthonormal coordinates( y , y ) and ˜ γ -orthonormal coordinates (˜ y , ˜ y ) with each centered at ζ . We choosethese coordinates so that at ζ we have ∂ y ∧ ∂ ˜ y = 0 = ∂ y ∧ ∂ ˜ y . Such coordinates can be constructed by fixing an auxiliary γ -orthonormal basis ( ∂ z , ∂ z ) of T ζ S ,observing that the matrix (cid:18) (cid:104) ∂ z , ∂ z (cid:105) ˜ γ (cid:104) ∂ z , ∂ z (cid:105) ˜ γ (cid:104) ∂ z , ∂ z (cid:105) ˜ γ (cid:104) ∂ z , ∂ z (cid:105) ˜ γ (cid:19) is symmetric, and hence has an orthonormal eigen-basis ( c c ) , ( c c ); defining ∂ y i = c ik ∂ z k yields a γ -orthonormal basis of T ζ S , and defining ∂ ˜ y i = (cid:107) ∂ y i (cid:107) − γ ∂ y i yieldsa ˜ γ -orthonormal basis of T ζ S . The coordinates ( y , y ) and (˜ y , ˜ y ) are respec-tively the γ and ˜ γ normal geodesic coordinates respectively associated to the frames( ∂ y , ∂ y ) and ( ∂ ˜ y , ∂ ˜ y ). For future use, we also note (cid:12)(cid:12) − (cid:107) ∂ y i (cid:107) γ (cid:12)(cid:12) ≤ (cid:15) by Lemma 4.5, and hence (cid:107) ∂ y i (cid:107) − γ ≤ (cid:12)(cid:12) − (cid:107) ∂ y i (cid:107) − γ (cid:12)(cid:12) ≤ (cid:15). (30)Next, we write ˜ γ = ˜ γ ik d ˜ y i ⊗ d ˜ y k and γ = γ ik dy i ⊗ dy k so that by evaluating at ζ we have ˜ γ = δ ik d ˜ y i ⊗ d ˜ y k and γ = δ ik dy i ⊗ dy k as wellas (cid:101) ∆( a ◦ ˜ u ) = tr( (cid:101) ∇ (cid:0) d ( a ◦ ˜ u )) (cid:1) = ˜ γ ik (cid:0) (cid:101) ∇ ∂ ˜ yi d ( a ◦ ˜ u ) (cid:1) ( ∂ ˜ y k ) − d ( a ◦ ˜ u )( (cid:101) ∇ ∂ ˜ yi ∂ ˜ y k ) (cid:1) = (cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ ˜ u ) (cid:1) ( ∂ ˜ y ) + (cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ ˜ u ) (cid:1) ( ∂ ˜ y )= (cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y ) + (cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y )+ (cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y ) + (cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y )and similarly ∆( a ◦ u ) = (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) + (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) . We now estimate (cid:12)(cid:12) ∆( a ◦ u ) − (cid:101) ∆( a ◦ ˜ u ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y ) (cid:12)(cid:12) . For the moment, let us write c i = (cid:107) ∂ y i (cid:107) − γ so that ∂ ˜ y i = c i ∂ y i (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) c (cid:101) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:0) c (cid:101) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) − c (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:0) c ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) + | − c | · (cid:12)(cid:12)(cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) ERAL CURVES AND MINIMAL SETS 69 ≤ c (cid:15) (cid:107) ∂ y (cid:107) γ + | − c | · (cid:12)(cid:12)(cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ c (cid:15) + | − c | · (cid:107) B (cid:107) γ ≤ (cid:15) + 2 (cid:15) (cid:107) B (cid:107) γ ≤ , where we have made use of the fact that (cid:12)(cid:12)(cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) = (cid:12)(cid:12) ∇ ∂ y (cid:0) d ( a ◦ u )( ∂ y ) (cid:1) − (cid:0) d ( a ◦ u ) (cid:1) ( ∇ ∂ y ∂ y ) (cid:12)(cid:12) = (cid:12)(cid:12) ∇ ∂ y (cid:0) d ( a ◦ u )( ∂ y ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ∇ ∂ y (cid:0) da ( u , ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ∇ u ,l (cid:0) da ( u , ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) ∇ u , da (cid:1) ( u , ) + da ( ∇ u , u , ) (cid:12)(cid:12) = (cid:12)(cid:12) da ( ∇ u , u , ) (cid:12)(cid:12) = (cid:12)(cid:12) da (cid:0) ( ∇ u , u , ) ⊥ (cid:1) + da (cid:0) ( ∇ u , u , ) (cid:62) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) da (cid:0) ( ∇ u , u , ) ⊥ (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) da (cid:0) B ( u , , u , ) (cid:1)(cid:12)(cid:12) ≤ (cid:107) B (cid:107) γ where we have used Corollary A.3 from Appendix A.1 which guarantees that ∇ da =0. We note that a similar estimate establishes that (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y d ( a ◦ u ) (cid:1) ( ∂ ˜ y ) − (cid:0) ∇ ∂ y d ( a ◦ u ) (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ . Next we note that (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y ) (cid:12)(cid:12) = c (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ y df (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:0) ∇ ∂ y df (cid:1) ( ∂ y ) (cid:12)(cid:12) + c (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ y df (cid:1) ( ∂ y ) − (cid:0) ∇ ∂ y df (cid:1) ( ∂ y ) (cid:12)(cid:12) ≤ c (cid:107)∇ df (cid:107) γ + (cid:15)c (cid:107) df (cid:107) γ ≤ . Similarly (cid:12)(cid:12)(cid:0) (cid:101) ∇ ∂ ˜ y df (cid:1) ( ∂ ˜ y ) (cid:12)(cid:12) ≤ . We conclude from these estimates that (cid:12)(cid:12) ∆( a ◦ u ) − (cid:101) ∆( a ◦ ˜ u ) (cid:12)(cid:12) ≤ . Combining these inequalities, we then find (cid:107) d ˜ α (cid:107) ˜ γ = | (cid:101) ∆( a ◦ ˜ u ) | by equation (27) , ≤ | ∆( a ◦ u ) | + 10 by above inequalities,= (cid:107) − d (( u ∗ da ) ◦ j ) (cid:107) γ + 10 by equation (27) , = (cid:107) − d ( u ∗ ( da ◦ J )) (cid:107) γ + 10 since u is J -holomorphic,= (cid:107) u ∗ dλ (cid:107) γ + 10 by Lemma 2 . , ≤ C h by Lemma 4.9 . This is the desired estimate, and hence we have completed the proof of Lemma4.12. (cid:3)
Core proofs.
With the elementary estimates established, we now move on toproving the main technical results, specifically Theorem 9, from which Theorem3 follows as an immediate corollary. We begin with a special case of the co-areaformula.
Lemma 4.13 (co-area formula with ˜ α ) . Let ( W, g ) be a smooth Riemannian manifold, and suppose S is a two dimensionalmanifold equipped with a smooth almost complex structure ˜ . To be clear, we require ∂S = ∅ . Suppose ˜ u : S → W satisfies ˜ u ∗ g ( x, y ) = ˜ u ∗ g (˜ x, ˜ y ) for all x, y ∈ T S . Let a : W → R be a smooth function, and ˜ α the one-form on S defined by ˜ α = − (˜ u ∗ da ) ◦ ˜ . Finally, we assume a ◦ ˜ u ( S ) ⊂ [ a , a ] . Then (cid:90) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) a a (cid:16) (cid:90) ( a ◦ ˜ u ) − ( t ) \X ˜ α (cid:17) dt, where X := { ζ ∈ S : d ( a ◦ ˜ u ) ζ = 0 } .Proof. We begin by defining (cid:101) S := S \ X and making a few observations. First, X isclosed and hence (cid:101) S ⊂ S is open, and therefore it carries the structure of a smoothmanifold. Second, by definition we have (˜ u ∗ da ) ∧ ˜ α (cid:12)(cid:12) X ≡
0, so (cid:90) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α. Observe that ˜ u : (cid:101) S → R × M is an immersion, and hence may be equipped withthe metric ˜ γ = ˜ u ∗ g . The almost complex structure ˜ on S induces an orientationon (cid:101) S , and hence we have(31) (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) (cid:101) S (cid:0) (˜ u ∗ da ) ∧ ˜ α (cid:1) ( ν, τ ) dµ γ , where ( ν, τ ) is a positively oriented ˜ γ -orthonormal frame field, and dµ γ is the volumeform on (cid:101) S associated to the metric ˜ γ ; see Section A.1 for further details. Equation(31) holds for arbitrary orthonormal frame field ( ν, τ ), however we shall henceforthmake use of the following particular frame. ν := (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) (cid:101) ∇ a ◦ ˜ u (cid:107) ˜ γ and τ := ˜ ν. Making use of the fact that ˜ is a ˜ γ -isometry and an almost complex structure, itis straightforward to verify the following.˜ α ( ν ) = 0 = ˜ u ∗ da ( τ )0 < ˜ u ∗ da ( ν ) = ˜ α ( τ )1 = (cid:107) τ (cid:107) γ = (cid:107) ν (cid:107) γ Also,(32) (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) ˜ γ = sup x ∈ T ζ S (cid:107) x (cid:107) ˜ γ =1 d ( a ◦ ˜ u )( x ) = sup x ∈ T ζ S (cid:107) x (cid:107) ˜ γ =1 da ( T ˜ u · x ) = ˜ u ∗ da ( ν ) , and(33) (cid:107) ˜ α (cid:107) ˜ γ = ˜ α ( τ ) = − ˜ u ∗ da (˜ ˜ ν ) = (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) ˜ γ . ERAL CURVES AND MINIMAL SETS 71
With ( ν, τ ) defined as such, we have the following. (cid:90) (cid:101) S (cid:0) (˜ u ∗ da ) ∧ ˜ α (cid:1) ( ν, τ ) dµ γ = (cid:90) (cid:101) S da ( T u · ν )˜ α ( τ ) dµ γ = (cid:90) (cid:101) S (cid:107) (cid:101) ∇ a ◦ ˜ u (cid:107) γ dµ γ Next we recall the following version of the co-area formula. A proof is provided inSection A.2.
Proposition 4.14 (The co-area formula) . Let ( S, γ ) be a C oriented Riemannian manifold of dimension two; we allow that S need not be complete . Suppose that β : S → [ a, b ] ⊂ R is a C function withoutcritical points. Let f : S → [0 , ∞ ) be a measurable function with respect to dµ γ .Then (34) (cid:90) S f (cid:107)∇ β (cid:107) γ dµ γ = (cid:90) ba (cid:16) (cid:90) β − ( t ) f dµ γ (cid:17) dt where ∇ β is the gradient of β computed with respect to the metric γ . We employ this result on (cid:101) S where γ = ˜ γ , β = a ◦ ˜ u , and f = (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) ˜ γ toobtain (cid:90) (cid:101) S (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) γ dµ γ = (cid:90) a a (cid:16) (cid:90) ( a ◦ ˜ u ) − ( t ) \X (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) ˜ γ dµ γ (cid:17) dt = (cid:90) a a (cid:16) (cid:90) ( a ◦ ˜ u ) − ( t ) \X ˜ α ( τ ) dµ γ (cid:17) dt = (cid:90) a a (cid:16) (cid:90) ( a ◦ ˜ u ) − ( t ) \X ˜ α (cid:17) dt, and hence by combining equalities we have (cid:90) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) a a (cid:16) (cid:90) ( a ◦ ˜ u ) − ( t ) \X ˜ α (cid:17) dt, which is the desired equality. This completes the proof of Lemma 4.13. (cid:3) We are now prepared to state and proof the main result of this section. Whilerather technical in its statement, it is applicable throughout the remainder of thismanuscript without need of generalization. A more accessible corollary is statedimmediately afterwards.
Theorem 9 (area bound estimate) . Let ( M, η = ( λ, ω )) be a framed Hamiltonian manifold, let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M , let C h be the ambient geometry constantgiven in Definition 4.11, and let E > be a positive constant, Then for each r > and tract of perturbed pseudoholomorphic map (˜ u, (cid:101) S, ˜ , f, u, S, j ) , satisfying That is, there may exist Cauchy sequences, with respect to g , which do not converge in S . Here we mean a tract of perturbed pseudoholomorphic map in the sense of Definition 4.2. (1) (cid:107) df (cid:107) γ + (cid:107)∇ df (cid:107) γ ≤ (cid:15) (1 + (cid:107) B u (cid:107) γ ) , where < (cid:15) < min(2 − , (1 + sup ζ ∈ supp( f ) (cid:107) B u ( ζ ) (cid:107) γ ) − ) , and (cid:107) df (cid:107) γ , (cid:107)∇ df (cid:107) γ , and (cid:107) B u (cid:107) γ are the L ∞ norms over the support of f ,(2) a ◦ ˜ u ( (cid:101) S ) ⊂ [0 , r ] ,(3) (0 , r ) ∩ a ◦ ˜ u ( ∂ (cid:101) S ) = ∅ ,(4) (cid:82) (cid:101) S u ∗ ω ≤ E ,(5) and r are regular values of a ◦ ˜ u ,we have (35) (cid:90) ( a ◦ ˜ u ) − ( r ) ˜ α ≤ (cid:16) C h E + (cid:90) ( a ◦ ˜ u ) − (0) ˜ α (cid:17) e C h r , and (36) (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω ≤ (cid:16) C − h (cid:90) ( a ◦ ˜ u ) − (0) ˜ α + E (cid:17)(cid:0) e C h r −
1) + E . Similarly for each r > and tract of perturbed pseudoholomorphic map, (˜ u, (cid:101) S, ˜ , f, u, S, j ) ,for which(1) a ◦ ˜ u ( (cid:101) S ) ⊂ [ − r, ,(2) ( − r, ∩ a ◦ ˜ u ( ∂ (cid:101) S ) = ∅ ,(3) (cid:82) (cid:101) S u ∗ ω ≤ E ,(4) and − r are regular values of a ◦ ˜ u ,we have (37) (cid:90) ( a ◦ ˜ u ) − ( − r ) ˜ α ≤ (cid:16) C h E + (cid:90) ( a ◦ ˜ u ) − (0) ˜ α (cid:17) e C h r , and inequality (36) again holds.Proof. Observe that the above problem has two distinct cases: the positive caseand the negative case; we refer to each as such. We also pause to recall someof the geometry involved. First, because (˜ u, (cid:101) S, ˜ , f, u, S, j ) is a tract of perturbedpseudoholomorphic curve in the sense of Definition 4.2, it follows that ˜ u : (cid:101) S → R × M is proper, and because M is compact and ˜ u ( (cid:101) S ) ⊂ [ − r, r ] × M , it follows that (cid:101) S iscompact. Second, the boundary of (cid:101) S , if it is not empty, is piecewise smooth, andcan be written as the union of two sets, namely ∂ (cid:101) S and ∂ (cid:101) S where the connectedcomponents of the former are level sets of a ◦ ˜ u , the latter are integral curves of (cid:101) ∇ ( a ◦ ˜ u ), and the set ∂ (cid:101) S ∩ ∂ (cid:101) S is finite and consists of precisely those non-smoothpoints of ∂ (cid:101) S .To proceed with the proof, we begin by fixing δ >
0. For the positive case, wedefine R + ⊂ [0 , r ] to be the set of regular values of the function a ◦ ˜ u : (cid:101) S → R .In the negative case we define R − ⊂ [0 , r ] to be the set of regular values of thefunction − a ◦ ˜ u : (cid:101) S → R . Depending on the case, we then define the followingfunctions. h ± : R ± ⊂ [0 , r ] → [0 , ∞ ) given by h ± ( s ) := (cid:90) ( a ◦ ˜ u ) − ( ± s ) ˜ α. ERAL CURVES AND MINIMAL SETS 73
Note that R ± are relatively open subsets of [0 , r ], and by Sard’s theorem they eachare of full measure in [0 , r ]. Recall that any open set of R can be written as thecountable union of disjoint open intervals, and hence we may write R ± = ∪ k ∈ N O ± k with the O ± k relatively open and pairwise disjoint. Henceforth, we will only considerthe case that O ± k (cid:54) = ∅ for all k ∈ N . More generally, one can always assume O ± k is never the empty set, however in such a case there may only be finitely manysuch open sets O ± k ; the proof in the finite case however is easily adapted from theinfinite case.Without loss of generality, we may re-index the {O ± k } k ∈ N to guarantee that 0 ∈O ± , and r ∈ O ± . Next, using the fact that each O ± k is a non-empty open interval,we may choose a sequence of finite sets of closed intervals {I ± ,n , I ± ,n , . . . , I ± n,n } n ∈ N with the following properties.( I I ± k,n = [ a ± k,n , b ± k,n ]( I
2) 0 = a ± ,n < b ± ,n < a ± ,n < b ± ,n < · · · < a ± n,n < b ± n,n = r ( I
3) for each n ∈ N we have ∪ nk =0 I ± k,n ⊂ ∪ n +1 k =0 I ± k,n +1 ( I r = lim n →∞ n (cid:88) k =0 | b ± k,n − a ± k,n | = lim n →∞ n (cid:88) k =0 ( b ± k,n − a ± k,n ) . Recall that because (cid:101) S is compact, it follows that (cid:82) (cid:101) S (˜ u ∗ da ) ∧ ˜ α < ∞ . By Lemma4.13, we then find that in the positive case we have (cid:90) r (cid:16) (cid:90) ( a ◦ ˜ u ) − ( s ) \X ˜ α (cid:17) ds = (cid:90) (cid:101) S \ ∂ (cid:101) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α < ∞ , where X = { ζ ∈ (cid:101) S : d ( a ◦ ˜ u ) ζ = 0 } ; here we have also made use of the factthat ∂ (cid:101) S ⊂ (cid:101) S has zero measure – this follows from the fact that ∂ (cid:101) S ⊂ (cid:101) S is apiecewise smooth embedded submanifold of codimension one. Also, the orientationon ( a ◦ ˜ u ) − ( s ) \ X is such that ˜ (cid:101) ∇ ( a ◦ ˜ u ) is a positive frame field. Similarly in thenegative case we have (cid:90) − r (cid:16) (cid:90) ( a ◦ ˜ u ) − ( s ) \X ˜ α (cid:17) ds = (cid:90) (cid:101) S \ ∂ (cid:101) S (˜ u ∗ da ) ∧ ˜ α = (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α < ∞ . Recall that ˜ α = − ˜ u ∗ da ◦ ˜ and ( v, ˜ v ) is a positively oriented basis for v (cid:54) = 0, andhence ˜ u ∗ da ∧ ˜ α is a non-negative function multiple of the area form on (cid:101) S associatedto ˜ γ , and hence the functions ˜ h ± ( s ) := (cid:82) ( a ◦ ˜ u ) − ( ± s ) \X ˜ α are integrable. Recall aconsequence of the dominated convergence theorem: if { A k } k ∈ N is a sequence ofsets A k ⊂ [ − r, r ] satisfying A k +1 ⊂ A k for all k ∈ N and the measure of the A k tends to zero as k → ∞ , then (cid:82) A k ˜ h ( s ) ds →
0. Again employing Lemma 4.13together with this latter application of the dominated convergence theorem we findthat 0 = lim n →∞ n − (cid:88) k =0 (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α, where(38) (cid:101) S a a = { ζ ∈ (cid:101) S : a ≤ a ◦ ˜ u ( ζ ) ≤ a } . Consequently, there exists an n ∈ N with the property that(39) C h e C h r n − (cid:88) k =0 (cid:16) (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α (cid:17) < δ with δ > n is fixed sufficiently large so that (39) holds.We now aim to study the growth rate of h ± . Recalling the definition of R ± andthe fact that the R ± are open, we see that h ± is smooth on R ± , and hence we willestimate | ( h ± ) (cid:48) | . To do this, it will be convenient to have first made the followingdefinitions. G + : R + → [0 , ∞ ) given by G + ( s ) := (cid:90) S s ˜ u ∗ ω.G − : R − → [0 , ∞ ) given by G − ( s ) := (cid:90) S − s ˜ u ∗ ω. Using the fact that ω evaluates non-negatively on complex lines, it follows that u ∗ ω evaluates non-negatively on positively oriented bases; then by definition of ˜ u ,particularly property (p4) of Definition 4.1, together with the fact that ω ( ∂ a , · ) ≡ u ∗ ω = ˜ u ∗ ω ; these two results together then show that the G ± aremonotone increasing, and since they are differentiable, we must have ( G ± ) (cid:48) ≥ R ± are open, we assume s ∈ R ± \ { r } ; then | ( h + ) (cid:48) ( s ) | = (cid:12)(cid:12)(cid:12) lim (cid:15) → + (cid:15) − (cid:0) h + ( s + (cid:15) ) − h + ( s ) (cid:1)(cid:12)(cid:12)(cid:12) = lim (cid:15) → + (cid:15) − (cid:12)(cid:12)(cid:12) (cid:90) ( a ◦ ˜ u ) − ( s + (cid:15) ) ˜ α − (cid:90) ( a ◦ ˜ u ) − ( s ) ˜ α (cid:12)(cid:12)(cid:12) = lim (cid:15) → + (cid:15) − (cid:12)(cid:12)(cid:12) (cid:90) (cid:101) S s + (cid:15)s d ˜ α (cid:12)(cid:12)(cid:12) = lim (cid:15) → + (cid:15) − (cid:12)(cid:12)(cid:12) (cid:90) (cid:101) S s + (cid:15)s \X d ˜ α (cid:12)(cid:12)(cid:12) ≤ lim (cid:15) → + (cid:15) − (cid:90) (cid:101) S s + (cid:15)s \X (cid:107) d ˜ α (cid:107) ˜ γ dµ γ ≤ lim (cid:15) → + (cid:15) − C h (cid:90) (cid:101) S s + (cid:15)s \X dµ γ ≤ C h lim (cid:15) → + (cid:15) − (cid:90) (cid:101) S s + (cid:15)s (cid:0) (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω (cid:1) = C h (cid:16) lim (cid:15) → + (cid:15) − (cid:90) (cid:101) S s + (cid:15)s (˜ u ∗ da ) ∧ ˜ α + lim (cid:15) → + (cid:15) − (cid:90) (cid:101) S s + (cid:15)s ˜ u ∗ ω (cid:17) = C h (cid:16) (cid:90) ( a ◦ u ) − ( s ) ˜ α + ( G + ) (cid:48) ( s ) (cid:17) = C h (cid:0) h + ( s ) + ( G + ) (cid:48) ( s ) (cid:1) , where to obtain the third equality we have made use of Stokes’ theorem and Lemma4.4, to obtain the second inequality we have employed Lemma 4.12, to obtain thethird inequality we have employed Lemma 4.8, and to obtain the sixth equality we ERAL CURVES AND MINIMAL SETS 75 have employed Lemma 4.13. A similar computation shows that | ( h − ) (cid:48) ( s ) | ≤ C h (cid:0) h − ( s ) + ( G − ) (cid:48) ( s ) (cid:1) . Summarizing, we have shown that for s ∈ R ± , we have the following differentialinequalities.(40) ( h ± ) (cid:48) ( s ) ≤ C h (cid:0) h ± ( s ) + ( G ± ) (cid:48) ( s ) (cid:1) Assume that a ± k,n and b ± k,n are as in ( I a ± k,n , b ± k,n ] ⊂ R ± ; then integrateinequality (40) on [ a ± k,n , s ] ⊂ [ a ± k,n , b ± k,n ] to obtain the following. h ± ( s ) ≤ h ± ( a ± k,n ) + C h (cid:90) sa ± k,n h ± ( t ) dt + C h (cid:0) G + ( s ) − G + ( a + k,n ) (cid:1) ≤ h ± ( a ± k,n ) + C h (cid:90) (cid:101) S b ± k,na ± k,n u ∗ ω + C h (cid:90) sa ± k,n h ± ( t ) dt, or in short,(41) h ± ( s ) ≤ h ± ( a ± k,n ) + C h (cid:90) (cid:101) S b ± k,na ± k,n u ∗ ω + C h (cid:90) sa ± k,n h ± ( t ) dt, for s ∈ [ a + k,n , b + k,n ]. In order to obtain a sharper estimate on h ± , we now employGronwall’s inequality. Lemma 4.15 (Gronwall’s inequality) . Assume that for t < t , the functions φ, ψ : [ t , t ] → [0 , ∞ ) are continuous.Suppose further that δ > and δ > are positive constants, and the followingestimate is satisfied for all t ∈ [ t , t ] φ ( t ) ≤ δ (cid:90) tt ψ ( t ) φ ( t ) ds + δ . Then for each t ∈ [ t , t ] the following estimate also holds. φ ( t ) ≤ δ e δ (cid:82) tt ψ ( s ) ds Proof.
See Section 1.3 of [38]. (cid:3)
Applying Lemma 4.15 to inequality (41) then yields h ± ( s ) ≤ (cid:16) h ± ( a ± k,n ) + C h E b ± k,n a ± k,n (cid:17) e C h ( s − a ± k,n ) = h ± ( a ± k,n ) e C h ( s − a ± k,n ) + C h E b ± k,n a ± k,n e C h ( s − a ± k,n ) (42)where we have abbreviated E a a = (cid:90) (cid:101) S a a u ∗ ω. Assume k < n and estimate h ( a ± k +1 ,n ) as follows, again making use of Lemma 4.4and Lemma 4.12, we find h ± ( a ± k +1 ,n ) = h ± ( b ± k,n ) + (cid:90) (cid:101) S a ± k +1 ,nb ± k,n d ˜ α ≤ h ± ( b ± k,n ) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n dµ γ ≤ h ± ( b ± k,n ) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n u ∗ ω = h ± ( b ± k,n ) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α + C h E a ± k +1 ,n b ± k,n . Making use of estimate (42), we have(43) h ± ( b ± k,n ) ≤ h ± ( a ± k,n ) e C h ( b ± k,n − a ± k,n ) + C h E b ± k,n a ± k,n e C h ( b ± k,n − a ± k,n ) h ± ( a ± k +1 ,n ) ≤ h ± ( b ± k,n ) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α + C h E a ± k +1 ,n b ± k,n . and thus h ± ( a ± k +1 ,n ) ≤ (cid:16) h ± ( a + k,n ) e C h ( b ± k,n − a ± k,n ) + C h E b ± k,n a ± k,n e C h ( b ± k,n − a ± k,n ) (cid:17) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α + C h E a ± k +1 ,n b ± k,n ≤ h ± ( a ± k,n ) e C h ( a ± k +1 ,n − a ± k,n ) + C h E a ± k +1 ,n a ± k,n e C h ( a ± k +1 ,n − a ± k,n ) + C h (cid:90) (cid:101) S a ± k +1 ,nb ± k,n (˜ u ∗ da ) ∧ ˜ α. Iterating this estimate backwards in k from n − h ± ( a ± n,n ) ≤ h ± ( a ± ,n ) e C h ( a + n,n − a ± ,n ) + C h E a ± n,n a ± ,n e C h ( a ± n,n − a ± ,n ) + C h e C h ( a + n,n − a ± ,n ) n (cid:88) (cid:96) =1 (cid:90) (cid:101) S a ± (cid:96),nb ± (cid:96) − ,n (˜ u ∗ da ) ∧ ˜ α. We recall that by construction a ± ,n = 0 and b ± n,n = r , so that with final applicationof estimate (42) we find h ± ( r ) ≤ (cid:16) h ± (0) + C h (cid:90) (cid:101) S u ∗ ω (cid:17) e C h r + C h e C h r n (cid:88) (cid:96) =1 (cid:90) (cid:101) S a ± (cid:96),nb ± (cid:96) − ,n (˜ u ∗ da ) ∧ ˜ α ≤ (cid:16) h ± (0) + C h (cid:90) (cid:101) S u ∗ ω (cid:17) e C h r + δ. Since δ > h ± ( r ) = (cid:90) ( a ◦ ˜ u ) − ( ± r ) ˜ α ≤ (cid:16) (cid:90) ( a ◦ u ) − (0) ˜ α + C h (cid:90) S u ∗ ω (cid:17) e C h r , which establishes inequalities (35) and (37).To achieve inequality (36) in both the positive and negative case, we essentiallyemploy Lemma 4.13 and observe that the functions s (cid:55)→ (cid:82) ( a ◦ u ) − ( ± s ) ˜ α is definedalmost everywhere (specifically on R ± which has full measure, and on R ± it agrees ERAL CURVES AND MINIMAL SETS 77 with h ± ); the result is then obtained by integrating the estimates just obtained for h ± ( r ). Indeed, in the positive case we have (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω = (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α + (cid:90) (cid:101) S u ∗ ω = (cid:90) r (cid:16) (cid:90) ( a ◦ ˜ u ) − ( s ) \X ˜ α (cid:17) ds + (cid:90) (cid:101) S u ∗ ω = (cid:90) r h + ( s ) ds + (cid:90) (cid:101) S u ∗ ω ≤ (cid:90) r (cid:16) h + (0) + C h E (cid:17) e C h s ds + (cid:90) (cid:101) S u ∗ ω ≤ (cid:16) C − h h + (0) + E (cid:17)(cid:0) e C h r −
1) + E , which establishes inequality (36) in the positive case; the negative case is establishedsimilarly. This completes the proof of Theorem 9. (cid:3) To complete Section 4.1, it remains to prove Theorem 3. This will follow as animmediate corollary to the following result.
Proposition 4.16 (area bounds more carefully) . Fix positive constants C H > , r > , and E > . Then there exists a con-stant C A = C A ( C H , r, E ) with the following significance. For each closed odd-dimensional manifold M equipped with the quadruple h = ( J, g, λ, ω ) where η :=( λ, ω ) is a Hamiltonian structure on M and ( J, g ) is an η -adapted almost Hermit-ian structure on R × M with the property that C h ≤ C H , where C h is the ambientgeometry constant established in Definition 4.11, and for each proper pseudoholo-morphic map u : S → I × M , where I = ( a − r, a + r ) , for which ∂S = ∅ , u − ( { a } × M ) = ∅ , and (cid:90) S u ∗ ω ≤ E < ∞ , the following also holds: (cid:90) S u ∗ ( da ∧ λ + ω ) ≤ C A . Proof.
Fix M , h , and u as in the hypotheses. By translation invariance of J , g , λ ,and ω , we may assume with out loss of generality that a = 0. Observe that since( a ◦ u ) − (0) = ∅ it follows that 0 is a regular value of a ◦ u . Next, fix a sequenceof (cid:15) k > (cid:15) k → k → ∞ , and with the additional property that ± (cid:15) k are regular values of a ◦ u for all k ∈ N ; note that such choice is possible by Sard’stheorem. Then we have (cid:90) S u ∗ ( da ∧ λ + ω ) = lim k →∞ (cid:90) S r − (cid:15)k − r + (cid:15)k u ∗ ( da ∧ λ + ω )where S r − (cid:15) k − r + (cid:15) k is defined as in (38). We now intend to employ Theorem 9 in thecase that the perturbed pseudoholomorphic map is given by (˜ u, (cid:101) S, ˜ , f, u, S, j ) =( u, S r − (cid:15) k , j, , u, S, j ). In this case ˜ α = u ∗ λ , so that by inequality (36) we have (cid:90) S r − (cid:15)k u ∗ ( da ∧ λ + ω ) ≤ E e C h ( r − (cid:15) k ) ≤ E e C h r =: C A . The same inequality holds for S − r + (cid:15) k instead of S r − (cid:15) k , and summing the twoinequalities then yields the desired result. (cid:3) As remarked above, Theorem 3 follows from Proposition 4.16 as an immediatecorollary.4.1.4.
An area estimate in realized Hamiltonian homotopies.
Here we prove a slightmodification of Proposition 4.16, for the case of realized Hamiltonian homotopiesin the sense of Definition 2.9.
Theorem 8 (area bounds in realized Hamiltonian homotopy) . Fix positive constants C H > , r > , and E > . Then there exists a constant C A = C A ( C H , r, E ) with the following significance. Let I × M, (ˆ λ, ˆ ω )) denote arealized Hamiltonian homotopy in the sense of Definition 2.9, and let ( J, g ) be anadapted almost Hermitian structure in the sense of Definition 2.10 with C H := sup q ∈I× M (cid:107) d ˆ λ q (cid:107) g ≤ C H . For each proper pseudoholomorphic map u : S → I r × M , where I r = ( a − r, a + r ) ⊂ I for which ∂S = ∅ , u − ( { a } × M ) = ∅ , and (cid:90) S u ∗ ω ≤ E < ∞ , the following also holds: Area u ∗ g ( S ) = (cid:90) S u ∗ ( da ∧ ˆ λ + ˆ ω ) ≤ C A . Additionally, for any [ a , a ] ⊂ I and any compact pseudoholomorphic map u : S → [ a , a ] × M for which a and a are regular values of a ◦ u and u − (cid:0) { a , a }× M (cid:1) = ∂S , the following also hold: (cid:90) Γ a u ∗ λ ≤ (cid:16) C H E + (cid:90) Γ a u ∗ λ (cid:17) e C H ( a − a ) , and (cid:90) Γ a u ∗ λ ≤ (cid:16) C H E + (cid:90) Γ a u ∗ λ (cid:17) e C H ( a − a ) , where Γ a i = ( a ◦ u ) − ( a i ) for i ∈ { , } . Similarly, for (cid:96) := min i ∈{ , } (cid:110) (cid:90) Γ ai u ∗ λ (cid:111) we have Area u ∗ g ( S ) ≤ ( C − H (cid:96) + E )( e C H ( a − a ) −
1) + E . Proof.
The proof is essentially identical to the proof of Theorem 9 and Proposition4.16, except with the following essentially typographical changes. In Theorem 9,the perturbed map ˜ u simply becomes the unperturbed map u ; or equivalently theperturbing function f ≡
0. Similarly, the perturbed almost complex structure ˜ on the domain S is simply the unperturbed j . All references to λ and ω should ERAL CURVES AND MINIMAL SETS 79 respectively be replaced with references to ˆ λ and ˆ ω . The one-form ˜ α is nothingmore than ˜ α = u ∗ ˆ λ. Instances of C h should be replaced with C H . The proof then goes through un-changed. (cid:3) Proof of Theorem 4: ω -Energy Threshold. The primary purpose of thissection is to prove Theorem 4, which we restate below.
Theorem 4 ( ω -energy threshold) . Let ( M, η = ( λ, ω )) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adapted almost Hermitian structure on R × M . Also, fix positive constants r > ,and C g > . Then there exists a positive constant < (cid:126) = (cid:126) ( M, η, J, g, r, C g ) withthe following significance. Let { h k } k ∈ N be a sequence of quadruples ( J k , g k , λ k , ω k ) with the property that each η k = ( λ k , ω k ) is a Hamiltonian structure on M , andeach ( J k , g k ) is an η k -adapted almost Hermitian structure on R × M , and supposethat ( J k , g k , λ k , ω k ) → ( J, g, λ, ω ) in C ∞ as k → ∞ . Furthermore, fix a ∈ R , and let u k : S k → R × M be a sequence of compactconnected generally immersed pseudoholomorphic maps which satisfy the followingconditions:( (cid:126)
1) either a ◦ u k ( S k ) ⊂ [ a , ∞ ) or a ◦ u k ( S k ) ⊂ ( −∞ , a ] for all k ∈ N ( (cid:126) Genus( S k ) ≤ C g ( (cid:126) a ◦ u k ( ∂S k ) ∩ [ a − r, a + r ] = ∅ ( (cid:126) a ∈ a ◦ u k ( S k ) .Then for all sufficiently large k ∈ N we have (cid:90) S k u ∗ k ω k ≥ (cid:126) . Proof.
For clarity of proof, we will first argue the case in which h k = ( J k , g k , λ k , ω k ) = ( J, g, λ, ω )for all k ∈ N . We argue by contradiction, so suppose not. Then there exists asequence ( u k , S k , j k ) of pseudoholomorphic maps satisfying properties ( (cid:126)
1) - ( (cid:126) (cid:90) S k u ∗ k ω → . For simplicity we will assume that a ◦ u k ( S k ) ⊂ [ a , ∞ ); the other case is arguedidentically. Making use of translation invariance of J , we will also assume withoutloss of generality that a = 0. Next, for each k ∈ N , fix a regular value a (cid:48) k of a ◦ u k with the property that a (cid:48) k ∈ ( r, r ). Then define (cid:98) S k to be ( a ◦ u k ) − ([0 , a (cid:48) k ]), whichwe observe has non-trivial intersection with ( a ◦ u k ) − (0). By Theorem 3, it followsthat there exists a constant C A > u ∗ k g ( (cid:98) S k ) = (cid:90) (cid:98) S k u ∗ k ( da ∧ λ + ω ) ≤ C A . Consequently, the sequence of pseudoholomorphic maps ( u k , (cid:98) S k , j k ) has uniformlybounded area, uniformly bounded genus, and each has compact domain (cid:98) S k and isgenerally immersed, and furthermore they each satisfy ( a ◦ u k )( (cid:98) S k ) = [0 , a (cid:48) k ] and u k ( ∂ (cid:98) S k ) ∩ [0 , r ] × M = ∅ . We conclude from Theorem 2.36, namely target-localGromov compactness, that there exists a (cid:48)(cid:48) ∈ ( r, r ) with the property that afterpassing to a subsequence, the pseudoholomorphic curves ( u k , (cid:101) S k , j k ) defined by (cid:101) S k := ( a ◦ u k ) − ([0 , a (cid:48)(cid:48) ]) converge in a Gromov sense to a nodal pseudoholomorphiccurve which has the property that its image both intersects { } × M non-triviallyand is also contained in [0 , a (cid:48)(cid:48) ] × M , and that the image of the boundary of thelimit Riemann surface is contained in { a (cid:48)(cid:48) } × M .As a consequence of the definition of Gromov convergence, we may choose con-nected components of (cid:101) S k (still denoted as (cid:101) S k ) with the property that u k ( (cid:101) S k ) ∩{ } × M (cid:54) = ∅ for all k ∈ N and such that the sequence of generally immersed pseu-doholomorphic curves ( u k , (cid:101) S k , j k ) converges in a Gromov sense to a nodal pseudo-holomorphic curve ( u, S, j, D ) with the property that u ( S ) is connected and again u ( ∂S ) ⊂ { a (cid:48)(cid:48) } × M . For any ζ ∈ S , define S ζ to be the connected component of S which contains ζ . We claim that there must exist ζ ∈ S such that a ◦ u ( ζ ) = 0 and u : S ζ → R × M is not a constant map. To see this, we suppose not and derivea contradiction. Indeed, note that there must exist ζ ∈ S such that a ◦ u ( ζ ) = 0,and u ( S ) is connected, so that if it is the case that u : S ζ → R × M is constantfor every such ζ then it must be the case that u : S → R × M is a constant map.However, if u : S → R × M is constant, then the sequence ( u k , (cid:101) S k , j k ) is converg-ing to a constant map, which is only possible if for all sufficiently large k we have u k : (cid:101) S k → R × M is a constant map, which contradicts the fact that the ( u k , (cid:101) S k , j k )are each generally immersed. This contradiction establishes that there does indeedexist ζ ∈ S such that a ◦ u ( ζ ) = 0 and u : S ζ → R × M is not a constant map.At this point we observe that u : S ζ → R × M is not a constant, and since it isconnected it must instead be generally immersed. Furthermore, we have a ◦ u ( ζ ) =0, u ( S ζ ) ⊂ [0 , a (cid:48)(cid:48) ] × M , u ( ∂S ζ ) ⊂ { a (cid:48)(cid:48) } × M , and u ( S ζ ) is connected. Since (cid:82) (cid:101) S k u ∗ k ω →
0, we must also have (cid:82) S ζ u ∗ ω = 0, and since ω evaluates non-negativelyon J -complex lines, we conclude that there exists a trajectory β : R → M of theHamiltonian vector field X η such that u ( S ) ⊂ [0 , a (cid:48)(cid:48) ] × β ( R ). Note that ( s, t ) (cid:55)→ ( s, β ( t )) is a holomorphic map, and from this it follows that a ◦ u : S ζ → R is aharmonic function, and hence can have no interior minima unless a ◦ u : S ζ → R is aconstant map. An elementary fact from complex variables shows that if a ◦ u : S ζ → R is a constant map, then u : S ζ → [0 , a (cid:48)(cid:48) ] × β ( R ) ⊂ R × M is a constant map, whichis impossible since ( u, S, j ) is generally immersed. Consequently, a ◦ u : S ζ → R canhave no interior minima, however from the above properties of ( u, S, j ) we see that a ◦ u achieves its absolute minimum at ζ ∈ S \ ∂S . This is the desired contradictionwhich completes our proof in the case that ( J k , g k , λ k , ω k ) = ( J, g, λ, ω ) for all k ∈ N .To complete the proof Theorem 4 it remains to consider the more general case inwhich ( J k , g k , λ k , ω k ) → ( J, g, λ, ω ). We note that in this case the proof is identicalwith a single modification: Rather than citing Theorem 3 to obtain local areabounds, we instead cite the more general Proposition 4.16; see below. Note thatthe hypotheses of Proposition 4.16 are satisfied precisely because ( J k , g k , λ k , ω k ) → ( J, g, λ, ω ) in C ∞ . (cid:3) ERAL CURVES AND MINIMAL SETS 81
Proof of Theorem 5: Asymptotic Connected-Local Area Bound.
The main purpose of this section is to prove Theorem 5, which roughly statesthat for each given feral curve, there exists a large compact set in the symplectiza-tion with the property that outside that compact set, the area of each connectedcomponent of the portion of the image of that curve contained in ball of some smallradius is universally bounded. Indeed, the bound is 1. We note that the large com-pact set will depend upon the curve, but the radius of the ball does not. Indeed, theradius depends only on the ambient geometry of the symplectization; specificallythe framed Hamiltonian structure and the almost Hermitian structure.Unfortunately, the proof of this result is rather long and complicated, so wegive the basic idea here, and then later we will elaborate further. Consider a feralcurve, and for some very large a > u : (cid:101) S → R × M where (cid:101) S = u − (cid:0) [ a − (cid:15), a + (cid:15) ] (cid:1) for some very small (cid:15) >
0. Thinking of (cid:15) as incredibly small and fixed, but a large, generic, and replaced with a larger value if needed, we then are inclined tothink of u : (cid:101) S → R × M as a ribbon of pseudoholomorphic curve which is very thin,but also very, very, long, so that the area is quite large, and the area gets largerwithout bound as a → ∞ . For each ζ ∈ S we then define (cid:101) S (cid:15) ( ζ ) to be theconnected component of u − ( B (cid:15) ( u ( ζ )) which contains ζ , where B r ( p ) denotesthe metric ball in R × M centered at p and of radius r . The essential question toask next is then: For each ζ ∈ ( a ◦ u ) − ( a ), can the area of (cid:101) S (cid:15) ( ζ ) be arbitrarilylarge by making a larger if needed? We will show that the answer is no, and thatthe idea is to cut the ribbon into tracts of pseudoholomorphic curves of modestlength and small height so that the area of each such tract is modest and so that (cid:101) S (cid:15) ( ζ ) is completely contained in one of the tracts. Put another way, we aim topartition the lower ribbon boundary ( a ◦ u ) − ( a − (cid:15) ) so that the length of each ofthe corresponding intervals is small, and so that the end point of each interval canbe flowed up to the top ribbon boundary ( a ◦ u ) − ( a + (cid:15) ) via the gradient flow. Wethen cut our ribbon along these gradient flow lines. Assuming the ω -energy of theribbon is small, which can always be guaranteed by making a sufficiently large, itis elementary to show the tracts of curve extending from partition intervals havemodest area, so the argument establishing Theorem 5 is complete if we can showthat (cid:101) S (cid:15) ( ζ ) is contained in one such tract.Of course there are a variety of obstacles to be overcome, and the bulk of theseare related to guaranteeing the existence of the desired intervals, specifically estab-lishing that most points in the lower ribbon boundary can be gradient flowed tothe top ribbon boundary. Indeed, although we have described it as a ribbon, itmay be the case that our portion of curve is in fact is the disjoint union of disks,annuli, pairs of pants, or other more topologically complicated surfaces. Further-more one must deal with the possibility that u : (cid:101) S → R × M may not be immersed, a ◦ u : (cid:101) S → R × M may not be Morse, and that many flow lines initiating in thelower ribbon boundary have terminal points which are local interior maxima. All ofthese issues are addressed, however significant preliminaries are necessary. Indeed,in Section 4.3.1 we provide Definition 4.17 which is the basic tool used throughoutSection 4.3, and we establish a number of important properties. In Section 4.3.2 we establish a few miscellaneous results which will be referenced in the main proof.Finally, in Section 4.3.3 we will provide the complete proof of Theorem 5, howeverthe bulk of the technical work is established in Proposition 4.30, which is essentiallya special case of Theorem 5 and is also proved in this section.4.3.1. Strip Estimates.
The purpose of this section is to provide the notion of aperturbed pseudoholomorphic strip, see Definition 4.17 below, and to establish afew technical properties, which will be used in later sections.Here and throughout, we let (
M, η ) be a framed Hamiltonian manifold with η = ( λ, ω ), and let ( J, g ) be an η -adapted almost Hermitian structure on the sym-plectization R × M . As in the previous section, specifically in equation (25), as-sociated to any perturbed pseudoholomorphic map (˜ u, ˜ , f, u, S, j ), we define thesmooth one-form ˜ α := − (˜ u ∗ da ) ◦ ˜ . Definition 4.17 (perturbed pseudoholomorphic strip) . Let ( M, η ) be a framed Hamiltonian manifold, ( J, g ) an η -adapted almost Hermitianstructure on R × M . A perturbed pseudoholomorphic strip consists of the tuple (˜ u, (cid:101) S, ˜ , f, u, S, j ) , where • (˜ u, ˜ , f, u, S, j ) is a perturbed pseudoholomorphic map in the sense of Defi-nition 4.1, • (cid:101) S ⊂ S is a compact manifold with boundary and corners, it is homeomorphicto a disk, and it is determined by the data ( p, I , h − , h + ) where(e1) I ⊂ R is an closed interval of finite length(e2) p : I → S is a smooth map for which ˜ u ◦ p : I → R × M is anembedding, a ◦ ˜ u ◦ p : I → R is the constant map a ◦ ˜ u ◦ p = a , and p ∗ ˜ α = dt where t is the coordinate on I induced from I ⊂ R (e3) h ± : I → R are C functions for which h − < h + , and for each t ∈ I there exists a map q t : (cid:104) min (cid:0) , h − ( t ) (cid:1) , max (cid:0) , h + ( t ) (cid:1)(cid:105) → S satisfying dds q t ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q t ( s ) (cid:1) (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q t ( s ) (cid:1) (cid:107) γ and q t (0) = p ( t ) , in which case (cid:101) S is given by (cid:101) S = (cid:91) t ∈I q t (cid:0) [ h − ( t ) , h + ( t )] (cid:1) ; here ˜ γ = ˜ u ∗ g and (cid:101) ∇ denotes the gradient with respect to the metric ˜ u ∗ g . In the case h − and h + are constant functions, we say (˜ u, (cid:101) S, ˜ ) isa rectangular f -perturbed pseudoholomorphic strip.Given (˜ u, ˜ , f, u, S, j ) , we will refer to (˜ u, (cid:101) S, ˜ , f, u, S, j ) , as the perturbed pseudo-holomorphic strip determined by the data ( p, I , h − , h + ) . Remark 4.18 (strips vs tracts) . It is worth pointing out the differences between perturbed pseudoholomorphic strips
ERAL CURVES AND MINIMAL SETS 83 and tracts of perturbed pseudoholomorphic maps. Ignoring the perturbation mo-mentarily, we note that rectangular strips are in fact special cases of tracts of pseu-doholomorphic maps, with the additional property that S is rectangular (that is,homeomorphic to a compact disk, and the boundary is piecewise smooth with fournon-smooth points), and a ◦ u : S → R has no critical points. The more gen-eral notion of a pseudoholomorphic strip then allows for the possibility that a ◦ u restricted to ∂ S need not be a constant map, and hence is no longer a tract ofpseudoholomorphic map. Regarding perturbations, the main difference is that fortracts of perturbed pseudoholomorphic maps, the support of the perturbation mustbe contained in the interior whereas for strips the support may overlap with theboundary. The following result establishes two important facts. First, given a perturbedpseudoholomorphic map, one can construct a perturbed pseudoholomorphic stripfrom less stringent data ( p, I , h − , h + ) than that given in Definition 4.17. Second,associated to each perturbed pseudoholomorphic strip are coordinates ( s, t ) whichhave a variety of properties. We make these facts precise with the following. Lemma 4.19 (strip reparametrization) . Let ( M, η ) be a framed Hamiltonian manifold, let ( J, g ) be an η -adapted almost Her-mitian structure on R × M , and let (˜ u, ˜ , f, u, S, j ) be a perturbed pseudoholomorphicmap. Suppose the -tuple (ˆ p, (cid:98) I , ˆ h − , ˆ h + ) consists of the following data.(1) (cid:98) I is an closed interval of finite length(2) ˆ p : (cid:98) I → S is a smooth map with image which is contained in a level set of a ◦ ˜ u and disjoint from the critical points of a ◦ ˜ u ,(3) ˆ h − and ˆ h + satisfy property (e3) of Definition 4.17. Then there exists an interval
I ⊂ R , and diffeomorphism ψ : I → (cid:98) I with theproperty that ( p, I , h − , h + ) := (ˆ p ◦ ψ, I , ˆ h − ◦ ψ, ˆ h + ◦ ψ ) satisfy properties (e1)- (e3) of Definition 4.17, and hence define a perturbed pseudoholomorphic strip (˜ u, (cid:101) S, ˜ , f, u, S, j ) . Moreover the diffeomorphism φ : { ( s, t ) ∈ R : t ∈ I and h − ( t ) < s < h + ( t ) } → (cid:101) Sφ ( s, t ) = q t ( s ) satisfies the following properties.(h1) a ◦ ˜ u ◦ φ ( s, t ) = a + s (h2) (˜ u ◦ φ ) ∗ g = ˆ γ = ˆ γ ds ⊗ ds + ˆ γ dt ⊗ dt for smooth functions ˆ γ kk = ˆ γ kk ( s, t ) and ˆ γ ≥ (h3) φ ∗ ˜ α = (cid:96) ( s, t ) dt , with < (cid:96) ≤ ˆ γ and (cid:96) (0 , t ) ≡ (h4) ( φ ∗ j ) ∂ s = τ ( s, t ) ∂ t with τ > .Consequently, each perturbed pseudoholomorphic strip can be given coordinates ( s, t ) determined by the equation ( s ( ζ ) , t ( ζ )) = φ − ( ζ ) .Proof. We begin by observing that if there were t ∈ (cid:98) I for which ˆ p ∗ ˜ α ( t ) = 0, thenˆ p ( t ) is a critical point of a ◦ ˜ u . To see this, recall that since a ◦ ˜ u ◦ ˆ p = const, itfollows that d ( a ◦ ˜ u )( T ˆ p · ∂ t ) = 0; but then if 0 = ˆ p ∗ ˜ α ( t ), then0 = ˆ p ∗ ˜ α ( t ) = − d ( a ◦ ˜ u )(˜ · T ˆ p · ∂ t ) (cid:12)(cid:12) t , so that d ( a ◦ ˜ u )(ˆ p ( t )) = 0, which is impossible since the image of ˆ p is disjoint fromthe critical points of d ( a ◦ ˜ u ) by assumption. After possibly precomposing with an orientation reversing diffeomorphism of (cid:98) I , we may consequently assume thatˆ p ∗ ˜ α ( ∂ t ) >
0. We now define the interval I by I := (cid:104) , (cid:90) (cid:98) I ˆ p ∗ ˜ α (cid:105) ⊂ R and define the map ψ by ψ : I → (cid:98) I by ψ ( t ) := F − ( t ) where F ( t ) := (cid:90) t ˜ α ˆ p (ˆ t ) (ˆ p (cid:48) (ˆ t )) d ˆ t ;here and above the subscripts denote the points of evaluation. To see that ψ is thedesired diffeomorphism, it is sufficient to show that(44) ˜ α (cid:0) (ˆ p ◦ ψ ) (cid:48) ( t ) (cid:1) ≡ . To establish this equality, we define the function G ( t ) = ˜ α ˆ p ( t ) (ˆ p (cid:48) ( t ))so that F (cid:48) = G and ψ = F − . Next recall that t = F ( F − ( t )), and differentiatingwe find ( F − ) (cid:48) ( t ) = F (cid:48) ( F − ( t )) , and hence˜ α ˆ p ◦ ψ ( t ) (cid:0) ˆ p (cid:48) ψ ( t ) · ψ (cid:48) ( t ) (cid:1) = ˜ α ˆ p ◦ ψ ( t ) (cid:16) ˆ p (cid:48) ψ ( t ) · F (cid:48) ( F − ( t )) (cid:17) = ˜ α ˆ p ◦ ψ ( t ) (cid:16) ˆ p (cid:48) ψ ( t ) · G ( ψ ( t )) (cid:17) = 1 . This verifies equation (44). With ψ established, it is straightforward to show thatthe tuple ( p, I , h − , h + ) := (ˆ p ◦ ψ, I , ˆ h − ◦ ψ, ˆ h + ◦ ψ ) satisfies properties (e1) - (e3) ofDefinition 4.17, and hence all that remains is to establish properties (h1) - (h4).Next we note that property (h4) follows from properties (h1) - (h3), and property(h1) follows immediately from the definition of the diffeomorphism φ .To prove property (h2), we let subscripts denote partial differentiation, and thenˆ γ ( ∂ s , ∂ t ) = g ( T ˜ u · φ s , T ˜ u · φ t ) . However by construction, for each fixed s the map t (cid:55)→ φ ( s , t ) is contained in alevel set of a ◦ ˜ u , and the vector field φ s is parallel to (cid:101) ∇ ( a ◦ ˜ u ). The gradient isorthogonal to level sets, and hence φ t and φ s are ˜ γ -orthogonal. Consequently ∂ s and ∂ t are ˆ γ -orthogonal, and ˆ γ = ˆ γ ds + ˆ γ dt as claimed. To see ˆ γ ≥
1, wefirst note:ˆ γ = (cid:107) φ s (cid:107) γ = ( da ( T ˜ u · φ s ) (cid:1) + (cid:0) λ ( T ˜ u · φ s ) (cid:1) + ω ( T ˜ u · φ s , J · T ˜ u · φ s ) . By Lemma 2.6, we see that ω ( T ˜ u · φ s , J · T ˜ u · φ s ) ≥
0. Also recall that φ s = (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) γ , and hence ˆ γ ≥ ( da ( T ˜ u · φ s ) (cid:1) = (cid:16) d ( a ◦ ˜ u )( (cid:101) ∇ ( a ◦ ˜ u )) (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) γ (cid:17) = 1 . This establishes property (h2).
ERAL CURVES AND MINIMAL SETS 85
Finally, to establish property (h3), observe φ ∗ ˜ α ( ∂ s ) = − d ( a ◦ ˜ u ) (cid:0) ˜ (cid:101) ∇ ( a ◦ ˜ u ) (cid:1) / (cid:107) (cid:101) ∇ ( a ◦ ˜ u ) (cid:107) γ = 0since ˜ is an almost complex structure and a ˜ γ -isometry. Consequently φ ∗ ˜ α = (cid:96) ( s, t ) dt . Note that (cid:96) (0 , t ) = ˜ α ( φ t (0 , t )) = ˜ α ( p (cid:48) ( t )) = 1by definition of φ and property (e2) of Definition 4.17. We also note that (cid:96) vanishesprecisely at the critical points of a ◦ ˜ u , and by construction none of these arecontained in the region (cid:101) S by assumption.To complete the proof of property (h3) all that remains is to show that (cid:96) ≤ ˆ γ .We estimate as follows. (cid:96) = (cid:0) ˜ α ( φ t ) (cid:1) = (cid:0) d ( a ◦ ˜ u )(˜ φ t ) (cid:1) ≤ (cid:107) d ( a ◦ ˜ u ) (cid:107) γ (cid:107) ˜ φ t (cid:107) γ = (cid:107) da (cid:107) g (cid:107) φ t (cid:107) γ = (cid:107) φ t (cid:107) γ = ˆ γ . Since (cid:96) and γ are both positive functions, the desired result is then immediate.This proves property (h3), and hence completes the proof of Lemma 4.19. (cid:3) Remark 4.20 (strip coordinates) . One immediate consequence of Lemma 4.19 above, is that it guarantees the existenceof the carefully defined structure of a perturbed pseudoholomorphic strip from verylittle, but necessary, geometric data. A second consequence is that it immediatelyguarantees coordinates ( s, t ) on a perturbed pseudoholomorphic strip which turn outto be quite useful. In particular, in light of properties (h2) and (h3) above, we willhenceforth assume ˜ u ∗ g = ˜ γ = ˜ γ ds ⊗ ds + ˜ γ dt ⊗ dt ˜ γ ( s, t ) ≥ γ ( s, t ) ≥ (cid:96) ( s, t ) where ˜ α = (cid:96) ( s, t ) dt. Later we will attempt to provide a uniform lower bound for (cid:96) ; for the momentthough, we only know that this function is smooth and positive.
Lemma 4.21 below can be thought of as an analog of Theorem 9 for perturbedpseudoholomorphic strips which are not too tall. Roughly speaking, it guaranteesthat if the integral of u ∗ λ (or, more precisely, the integral of − ˜ u ∗ da ◦ ˜ along thetop boundary is more than twice the integral along the bottom boundary, the stripmust capture some ω -energy proportional to this difference. We note that in whatfollows we adapt our notation from Definition 4.2 in which ∂ (cid:101) S = ∂ (cid:101) S ∪ ∂ (cid:101) S , where ∂ (cid:101) S is the “top” and “bottom” boundaries of of (cid:101) S , and ∂ (cid:101) S is the “side” boundaryof (cid:101) S . Lemma 4.21 (general strip estimate) . Let ( M, η ) be a framed Hamiltonian manifold, let ( J, g ) be an η -adapted almostHermitian structure on R × M , let C h be the associated ambient geometry con-stant established in Definition 4.11, and let (˜ u, (cid:101) S, ˜ , f, u, S, j ) be a perturbed J -stripdetermined by the data ( p, I , h − , h + ) , with (45) − ln 22 C h ≤ h − ≤ ≤ h + ≤ ln 22 C h . With ( s, t ) the coordinates on (cid:101) S as guaranteed by Lemma 4.19, we define ∂ ± (cid:101) S := { ( s, t ) ∈ ∂ (cid:101) S : ± s > } , with orientation such that ˜ α defines a positive volume form on each. Then (cid:90) ∂ +0 (cid:101) S ˜ α − (cid:90) ∂ − (cid:101) S ˜ α ≤ C h (cid:90) (cid:101) S u ∗ ω, and similarly (cid:90) ∂ − (cid:101) S ˜ α − (cid:90) ∂ +0 (cid:101) S ˜ α ≤ C h (cid:90) (cid:101) S u ∗ ω. Proof.
Without loss of generality, I = [0 , b ]. For each k ∈ N and 1 ≤ (cid:96) ≤ k definethe interval I k,(cid:96) := (cid:2) (cid:96) − k b, (cid:96) k b (cid:3) . Similarly, define h + k,(cid:96) := inf t ∈I k,(cid:96) h + ( t ) and h − k,(cid:96) := sup t ∈I k,(cid:96) h − ( t ). By definition ofintegrability, we have (cid:90) { ( s,t ) ∈ (cid:101) S : s ≥ } d ˜ α = lim k →∞ k (cid:88) (cid:96) =1 (cid:90) [0 ,h + k,(cid:96) ] ×I k,(cid:96) d ˜ α and (cid:90) { ( s,t ) ∈ (cid:101) S : s ≤ } d ˜ α = lim k →∞ k (cid:88) (cid:96) =1 (cid:90) [ h − k,(cid:96) , ×I k,(cid:96) d ˜ α, and hence by Stokes’ theorem and the fact that ˜ α ( ∂ s ) = 0, we have (cid:90) ∂ +0 (cid:101) S α = lim k →∞ k (cid:88) (cid:96) =1 (cid:90) { h + k,(cid:96) }×I k,(cid:96) ˜ α, and (cid:90) ∂ − (cid:101) S α = lim k →∞ k (cid:88) (cid:96) =1 (cid:90) { h − k,(cid:96) }×I k,(cid:96) ˜ α. At this point we observe that restricting ˜ u to each [ h − k,(cid:96) , h + k,(cid:96) ] × I k,(cid:96) defines a rect-angular perturbed pseudoholomorphic strip, and hence by Theorem 9 that(46) (cid:90) { h + k,(cid:96) }×I k,(cid:96) ˜ α ≤ (cid:16) C h (cid:90) [ h − k,(cid:96) ,h + k,(cid:96) ] ×I k,(cid:96) u ∗ ω + (cid:90) { h − k,(cid:96) }×I k,(cid:96) ˜ α (cid:17) e C h ( h + k,(cid:96) − h − k,(cid:96) ) . Making use of the inequalities (45) and rearranging, we find(47) (cid:90) { h + k,(cid:96) }×I k,(cid:96) ˜ α − (cid:90) { h − k,(cid:96) }×I k,(cid:96) ˜ α ≤ C h (cid:90) [ h − k,(cid:96) ,h + k,(cid:96) ] ×I k,(cid:96) u ∗ ω. ERAL CURVES AND MINIMAL SETS 87
Summing and passing to the limit then yields (cid:90) ∂ +0 (cid:101) S ˜ α − (cid:90) ∂ − (cid:101) S ˜ α = lim k →∞ k (cid:88) (cid:96) =1 (cid:90) { h + k,(cid:96) }×I k,(cid:96) ˜ α − k →∞ k (cid:88) (cid:96) =1 (cid:90) { h − k,(cid:96) }×I k,(cid:96) ˜ α ≤ lim k →∞ k (cid:88) (cid:96) =1 C h (cid:90) [ h − k,(cid:96) ,h + k,(cid:96) ] ×I k,(cid:96) u ∗ ω ≤ C h (cid:90) (cid:101) S u ∗ ω. This is the desired estimate, and hence completes the proof of Lemma 4.21. (cid:3)
In light of Remark 4.20, and with the aid of Lemma 4.21, our next task is toattempt to obtain a uniform lower bound on the smooth positive function (cid:96) definedby the property that ˜ α = (cid:96) ( s, t ) dt , since doing so would be quite beneficial for laterestimates. Unfortunately, such a uniform pointwise estimate is not true, but ratherthe desired pointwise estimate holds everywhere except on a set which is small ina controlled manner. We make this precise with Lemma 4.22 below. Lemma 4.22 ( λ -shrinkage) . Let ( M, η ) be a framed Hamiltonian manifold and ( J, g ) be an η -adapted almostHermitian structure on R × M , and let (˜ u, (cid:101) S, ˜ , f, u, S, j ) be a perturbed pseudo-holomorphic strip determined by the data ( p, I , h − , h + ) , with coordinates ( s, t ) asguaranteed by Lemma 4.19. Let C h be the associated ambient geometry constantestablished in Definition 4.11, Suppose that for each t ∈ I we have − ln 22 C h ≤ h − ( t ) ≤ ≤ h + ( t ) ≤ ln 22 C h . Define the set K := (cid:8) t ∈ I : ≥ inf h − ( t ) ≤ s ≤ h + ( t ) (cid:96) ( s, t ) (cid:9) where ˜ α = (cid:96) ( s, t ) dt . Then C h (cid:90) (cid:101) S u ∗ ω ≥ µ ( K ) where µ is the Lebesgue measure associated to the coordinate t ∈ I .Proof. We begin by observing that K is compact. Next we define A := { ( s, t ) ∈ (cid:101) S : (cid:96) ( s, t ) ≤ } . Next, for each ζ = ( s , t ) ∈ A we define some quantities, which we also describegeometrically below: σ ( ζ ) = σ ( s , t ) = sup (cid:8) ˆ s ∈ [0 , h + ( t )] : inf ≤ s ≤ ˆ s { (cid:96) ( s, t ) } ≥ (cid:9) if s > (cid:8) ˆ s ∈ [ h − ( t ) ,
0] : inf ˆ s ≤ s ≤ { (cid:96) ( s, t ) } ≥ (cid:9) if s < x ( ζ ) = x ( s , t ) = inf (cid:8) τ ∈ I : τ ≤ t and sup τ ≤ t ≤ t { (cid:96) ( σ ( ζ ) , t ) } ≤ (cid:9) y ( ζ ) = y ( s , t ) = sup (cid:8) τ ∈ I : τ ≥ t and sup t ≤ t ≤ τ { (cid:96) ( σ ( ζ ) , t ) } ≤ (cid:9) . We take a moment to describe these functions in a more geometric context.First, recall that in the coordinates ( s, t ), the s coordinate can be thought of asthe symplectization coordinate, and the t coordinate measuring movement within a symplectization level set. Suppose we are given some ( s , t ) ∈ A with s > t = t , which by construction is an integral curveof (cid:101) ∇ ( a ◦ ˜ u ). In particular, we restrict the smooth function (cid:96) to this path, and notethat (cid:96) (0 , t ) = 1 by construction, so that as we increase s from zero, there must bea first value for which (cid:96) ( s, t ) = . This first such s is then defined to be σ ( s , t ),provided s is positive, which we have indeed assumed. A similar construction holdsfor the case that s is negative.To understand the functions x and y , we can consider the point ( s , t ) ∈ A ,again assuming s >
0. Recall that by definition of A , we have (cid:96) ( s , t ) ≤ .To the point ( s , t ) we will associate the point ( σ ( s , t ) , t ), and we recall that (cid:96) ( σ ( s , t ) , t ) = by construction, so that necessarily σ ( s , t ) < s ≤ h + ( t ) . (48)We then we aim to construct a path of the form { σ ( s , t ) } × I on which (cid:96) ≤ ;here I is an interval. This is possible since (cid:96) is continuous and (cid:96) ( σ ( s , t ) , t ) = < . Thus we define the functions x and y to respectively be the smallest andlargest possible values for which I := [ x ( s , t ) , y ( s , t )] has the property that on { σ ( s , t ) } × I we have (cid:96) ≤ . Moreover in light of inequality (48) it also followsthat x ( s , t ) < t < y ( s , t )whenever t ∈ I \ ∂ I , and for all t ∈ I we have x ( s , t ) < y ( s , t ).With the functions x and y defined and understood, we can now define thefollowing collection of (relatively) open subsets O ζ ⊂ I for each ζ = ( s , t ) ∈ A . O ζ := (cid:2) x ( ζ ) , y ( ζ ) (cid:1) if x ( ζ ) = t (cid:0) x ( ζ ) , y ( ζ ) (cid:3) if y ( ζ ) = t (cid:0) x ( ζ ) , y ( ζ ) (cid:1) otherwise.Observe that since (cid:96) is continuous and (cid:101) S is compact, it follows that if t ∈ K , thenthere exists s ∈ ( h − ( t ) , h + ( t )) such that ( s , t ) ∈ A , and by construction t ∈O ζ . Consequently {O ζ } ζ ∈K is an open cover of K , which as previously mentioned,is compact. It follows that there exists a finite set { ζ − , . . . , ζ − k − , ζ +1 , . . . , ζ + k + } ⊂ A such that K ⊂ (cid:0) k − (cid:91) i =1 O ζ − i (cid:1) ∪ (cid:0) k + (cid:91) i =1 O ζ + i (cid:1) , and for which ζ ± i = ( s ± i , t ± i ) with s + i ≥ s − i <
0. The pairwise intersection ofthese open intervals may be open and nonempty, so we refine this set of intervalsby the following inductive procedure.˜ U − := O ζ − ˜ U − i := O ζ − i \ ∪ i − i (cid:48) =1 ˜ U − i (cid:48) ˜ U +1 := O ζ +1 \ ∪ k − i (cid:48) =1 ˜ U − i (cid:48) ˜ U + i := O ζ + i \ (cid:16)(cid:0) ∪ k − i (cid:48) =1 ˜ U − i (cid:48) (cid:1) ∪ (cid:0) ∪ i − i (cid:48) =1 ˜ U + i (cid:48) (cid:1)(cid:17) , ERAL CURVES AND MINIMAL SETS 89 where the bar denotes closure. Observe that each ˜ U − i and ˜ U + i is the union of finitelymany disjoint open intervals, and hence we may further write˜ U − i = m − i (cid:91) i (cid:48) =1 U − i,i (cid:48) and ˜ U + i = m + i (cid:91) i (cid:48) =1 U + i,i (cid:48) , where each U − i,i (cid:48) and U + i,i (cid:48) is an open interval.Finally we define the following finite sets of products of closed intervals. S + i,i (cid:48) := [0 , s + i,i (cid:48) ] × U + i,i (cid:48) for 1 ≤ i ≤ k + with s + i,i (cid:48) := σ ( ζ + i ) ≥ , and 1 ≤ i (cid:48) ≤ m + i S − i,i (cid:48) := [ s − i,i (cid:48) , × U − i,i (cid:48) for 1 ≤ i ≤ k − with s − i,i (cid:48) := σ ( ζ − i ) < , and 1 ≤ i (cid:48) ≤ m − i For later computational clarity, it will be convenient to reindex these sets by ν ± : { , . . . , m ± := k ± (cid:88) i =1 m ± i } → { (1 , , . . . , (1 , m ± ) , (2 , , . . . , (2 , m ± ) , . . . , ( k, m ± k ) } so that S + ν + = [0 , s + ν + ] × U + ν + for ν + ∈ { , . . . , m + } and S − ν − = [ s − ν − , × U − ν − for ν − ∈ { , . . . , m − } . For convenience, define U := (cid:0) ∪ m − ν − =1 U − ν − (cid:1) (cid:91) (cid:0) ∪ m + ν + =1 U + ν + (cid:1) , so that { } × U = (cid:0) ( ∪ m − ν − =1 S − ν − ) ∪ ( ∪ m + ν + =1 S + ν + ) (cid:1) ∩ (cid:0) { } × I (cid:1) . The following facts arethen straightforward to verify.(S1) K ⊂ U (S2) inf t ∈I\U h − ( t ) ≤ s ≤ h + ( t ) { (cid:96) ( s, t ) } ≥ (S3) for each ν ± ∈ { , . . . , m ± } we havesup t ∈U ± ν ± (cid:96) ( s ± ν ± , t ) ≤ (S4) the pairwise intersection of elements of the set { S − , . . . , S − m − , S +1 , . . . , S + m + } have empty interior in (cid:101) S .Observe that since ˜ α = (cid:96) dt , and since (cid:96) (0 , t ) ≡
1, we may employ property (S3) toestimate (cid:90) { s ± ν ± }×U ± ν ± ˜ α = (cid:90) sup U ± ν ± inf U ± ν ± (cid:96) ( s ± ν ± , t ) dt ≤ (cid:90) sup U ± ν ± inf U ± ν ± dt = (cid:90) { }×U ± ν ± ˜ α. Observe that each (˜ u, S ± ν ± , ˜ , f, u, S, j ) is a rectangular perturbed pseudoholomor-phic strip determined by the data ( p, U ± ν ± , h − ν ± , h + ν ± ) where h − ν + = 0, h + ν + = s + ν + , h − ν − = s ν − , and h + ν − = 0. Moreover, we have | s ± ν ± | ≤ sup t | h ± ( t ) | ≤ ln 22 C h and (cid:90) { s ± ν ± }×U ± ν ± ˜ α ≤ (cid:90) { }×U ± ν ± ˜ α, and thus letting µ denote the Lebesgue measure associated to the coordinate t , weemploy Lemma 4.21 to estimates µ ( U ± ν ± ) = (cid:90) { }×U ± ν ± ˜ α = (cid:90) { }×U ± ν ± ˜ α − (cid:90) { }×U ± ν ± ˜ α ≤ (cid:90) { }×U ± ν ± ˜ α − (cid:90) { s ± ν ± }×U ± ν ± ˜ α ≤ C h (cid:90) S ± ν ± ˜ u ∗ ω. Or more concisely, µ ( U ± ν ± ) ≤ C h (cid:90) S ± ν ± ˜ u ∗ ω. Using property (S4) and the fact that ω evaluates non-negatively on complex lines,it follows that 4 C h (cid:90) (cid:101) S u ∗ ω ≥ C h m ± (cid:88) ν ± =1 (cid:90) S ± ν ± u ∗ ω ≥ m ± (cid:88) ν ± =1 µ ( U ± ν ± )= µ ( ∪ m ± ν ± =1 U ± ν ± ) = µ ( U ) ≥ µ ( K ) , where the final equality follows from property (S1). (cid:3) We finish Section 4.3.1 with Lemma 4.23 below, which roughly states that if apseudoholomorphic strip is not too “tall,” and the ω -energy is less than or equal tothe “height” times the “width” then there must exist a gradient trajectory goingfrom bottom to top with length with is not too long (relative to the height of thestrip). Lemma 4.23 (modest length flow lines) . Let ( M, η ) be a framed Hamiltonian manifold and ( J, g ) be an η -adapted almostHermitian structure on R × M . Let C h be the associated ambient geometry constantestablished in Definition 4.11. Let (˜ u, ˜ , f, u, S, j ) be a perturbed pseudoholomorphicmap, and fix (cid:15) ∈ R such that < (cid:15) < min(2 − , (1 + sup ζ ∈ supp( f ) (cid:107) B u ( ζ ) (cid:107) γ ) − ) . ERAL CURVES AND MINIMAL SETS 91
Suppose further that (cid:107) df (cid:107) γ + (cid:107)∇ df (cid:107) γ ≤ (cid:15) (1 + (cid:107) B u (cid:107) γ ) , where (cid:107) df (cid:107) γ , (cid:107)∇ df (cid:107) γ , and (cid:107) B u (cid:107) γ are the L ∞ norms over the support of f . Thenfor any finite set of rectangular perturbed pseudoholomorphic strips, denoted by { (˜ u k , (cid:101) S k , ˜ k , f, u, S, j ) } nk =1 , satisfying(R1) a = inf ζ ∈ (cid:101) S k a ◦ ˜ u k ( ζ ) , independent of k (R2) a = sup ζ ∈ (cid:101) S k a ◦ ˜ u k ( ζ ) , independent of k (R3) a − a ≤ C h (R4) (cid:80) nk =1 (cid:82) (cid:101) S k u ∗ k ω ≤ ( a − a ) (cid:80) nk =1 (cid:82) ( a ◦ ˜ u k ) − ( a ) ˜ α, (R5) (cid:101) S k ∩ (cid:101) S k (cid:48) = ∅ for k (cid:54) = k (cid:48) there exists k ∈ { , . . . , n } and a solution to the differential equation q : [0 , s ] → S k q (cid:48) ( s ) = (cid:101) ∇ ( a ◦ ˜ u k ) (cid:0) q ( s ) (cid:1) a (˜ u k ( q (0))) = a a (˜ u k ( q ( s ))) = a for which length ˜ γ (cid:0) q ([0 , s ]) (cid:1) ≤ ( a − a ) . Proof.
For convenience, we let (˜ u, (cid:101) S, ˜ , f, u, S, j ) denote the union of the rectangularperturbed pseudoholomorphic strips { (˜ u k , (cid:101) S k , ˜ k , f, u, S, j ) } nk =1 so that (cid:101) S = ∪ nk =1 (cid:101) S k and ˜ u (cid:12)(cid:12) (cid:101) S k = ˜ u k , and similarly for ˜ . Next we define the constants c := a − a and c := (cid:90) ( a ◦ ˜ u ) − ( a ) ˜ α, and equip (cid:101) S with coordinates ( s, t ) via Lemma 4.19 so that using these coordinatesto parameterize (cid:101) S we have (cid:101) S = [0 , c ] × I where I ⊂ R is the union of finitely manypairwise disjoint closed intervals with total length( I ) = c . Recall that anotherconsequence of Lemma 4.19, specifically property (h2), is that˜ γ = ˜ u ∗ g = ˜ γ ds + ˜ γ dt . Consequently, to prove Lemma 4.23, it is sufficient to prove(49) c := inf t ∈I (cid:90) c ˜ γ ( s, t ) ds ≤ c . Next we define the closed set K similarly to the way it was defined in Lemma 4.22: K := (cid:8) t ∈ I : ≥ inf ≤ s ≤ c (cid:96) ( s, t ) (cid:9) where ˜ α = (cid:96) ( s, t ) dt . Recall that as a consequence of Lemma 4.22 we have µ ( K ) ≤ C h (cid:82) (cid:101) S u ∗ ω , where µ denotes the Lebesgue measure on I associated to the coordi-nate t . We then make the following estimate.Area ˜ γ ( (cid:101) S ) = Area ˜ γ (cid:0) [0 , c ] × I (cid:1) = (cid:90) I (cid:90) c (cid:0) ˜ γ ˜ γ (cid:1) dsdt ≥ (cid:90) I (cid:90) c ˜ γ (cid:96) dsdt ≥ (cid:90) I\K (cid:90) c ˜ γ (cid:96) dsdt ≥ (cid:90) I\K (cid:90) c ˜ γ dsdt ≥ c (cid:90) I\K dt = c (cid:0) c − µ ( K ) (cid:1) ≥ c (cid:0) c − C h (cid:90) (cid:101) S u ∗ ω (cid:1) ;(50)where to obtain the second equality we have employed equation (137) from SectionA.1 which expresses the Hausdorff measure associated to a Riemannian metric inlocal coordinates, and we have used the fact that γ = γ = 0; to obtain thefirst inequality we have made use of the fact that γ ≥ (cid:96) which was establishedin property (h3) of Lemma 4.19; the third inequality makes use of the definitionof K ; the fourth inequality follows from the definition of c ; and the final equalityemploys Lemma 4.22. Note, we also have the following estimate. (cid:90) (cid:101) S d (cid:0) ( a ◦ ˜ u − a )˜ α (cid:1) + u ∗ ω = (cid:16) (cid:90) (cid:101) S (˜ u ∗ da ) ∧ ˜ α + ˜ u ∗ ω (cid:17) + (cid:90) (cid:101) S ( a ◦ ˜ u − a ) d ˜ α ≥ Area ˜ γ ( (cid:101) S ) + (cid:90) (cid:101) S ( a ◦ ˜ u − a ) d ˜ α ≥ Area ˜ γ ( (cid:101) S ) − c C h Area ˜ γ ( (cid:101) S )= ( − c C h )Area ˜ γ ( (cid:101) S ) , where the first inequality follows from Lemma 4.8, and the second inequality followsfrom Lemma 4.12. We now recall that ∂ (cid:101) S = ∂ (cid:101) S ∪ ∂ (cid:101) S with a ◦ ˜ u ( ∂ (cid:101) S ) = { a , a } and that ∂ (cid:101) S consists of integral curves of (cid:101) ∇ ( a ◦ ˜ u ). We also recall Lemma 4.4which guarantees that ˜ α ( (cid:101) ∇ ( a ◦ ˜ u )) ≡
0, and hence (cid:90) (cid:101) S d (cid:0) ( a ◦ ˜ u − a )˜ α (cid:1) = (cid:90) ( a ◦ ˜ u ) − ( a ) ( a ◦ ˜ u − a )˜ α − (cid:90) ( a ◦ ˜ u ) − ( a ) ( a ◦ ˜ u − a )˜ α + (cid:90) ∂ S ( a ◦ ˜ u − a )˜ α = 0 + c (cid:90) ( a ◦ ˜ u ) − ( a ) ˜ α + 0= c c . Combining the above inequalities then yields the following. c c + (cid:90) (cid:101) S u ∗ ω ≥ ( − c C h )Area ˜ γ ( (cid:101) S )Combining this with inequality (50), then yields the following.(51) c c + (cid:90) (cid:101) S u ∗ ω ≥ ( − c C h ) c ( c − C h (cid:90) (cid:101) S u ∗ ω ) . Finally, we recall our assumptions (R3) and (R4), which can be restated as c = a − a ≤ C h and (cid:90) (cid:101) S u ∗ ω ≤ c c . ERAL CURVES AND MINIMAL SETS 93
From these it is elementary to establish the following.2 c c ≥ c c + (cid:90) (cid:101) S u ∗ ω − c C h ≥ c − C h (cid:90) (cid:101) S u ∗ ω ≥ c . We now combine these inequalities with (51) to obtain c ≤ c which is the desired inequality as stated in (49). This completes the proof of Lemma4.23. (cid:3) Some preliminary miscellany.
The purpose of this Section is to establish afew miscellaneous results to be referenced later in the proof of Theorem 5. Firstly,these consist of the notion of a ( δ, (cid:15) )-tame perturbation of a pseudoholomorphiccurve, see Definition 4.24 below, which essentially provides a certain class of per-turbations which are sufficiently small so that a variety of estimates hold automat-ically, and then we establish that such perturbations exist in sufficient abundance;see Lemma 4.26. Secondly, we also establish that in a very particular measuretheoretic sense, tangent planes of pseudoholomorphic curves with small ω -energyhave tangent planes which are usually almost vertical; see Lemma 4.27 below. Andfinally, we establish the existence of a small geometric constant r which will bemade use of extensively in Section 4.3.3; see Lemma 4.29. Definition 4.24 (( δ, (cid:15) )-tame perturbations) . Let ( M, η ) be a framed Hamiltonian manifold, ( J, g ) be an η -adapted almost Her-mitian structure on the symplectization R × M , let (˜ u, ˜ , f, u, S, j ) be a perturbedpseudoholomorphic map, and let δ, (cid:15) > . We say (˜ u, ˜ , f, u, S, j ) is a ( δ, (cid:15) ) -tameperturbed pseudoholomorphic map provided the following hold, where Z = { ζ ∈ S | T u ( ζ ) = 0 } .(d1) δ < min (cid:16) dist γ (Crit a ◦ u , ∂S ) , min ζ ,ζ ∈Z ζ (cid:54) = ζ dist γ ( ζ , ζ ) , dist γ ( Z , ∂S ) (cid:17) (d2) (cid:15) < min(2 − , C B ) (d3) the restricted map f : S \ { ζ ∈ S : dist γ ( ζ, Z ) < δ } : → R is Morse(d4) sup ζ ∈ Ω | f ( ζ ) | + sup ζ ∈ Ω (cid:107) df ( ζ ) (cid:107) γ + sup ζ ∈ Ω (cid:107)∇ df ( ζ ) (cid:107) γ ≤ (cid:15) (1 + C B ) where Ω := supp( f ) , Crit a ◦ u is the set of critical points of a ◦ u : S → R , γ = u ∗ g , ∇ is covariant differentiation with respect to the Levi-Civita connection associatedto the metric γ , B u is the second fundamental form associated to u as recalled inDefinition A.4 of Section A.1, and C B := sup {(cid:107) B u ( ζ ) (cid:107) γ : dist γ ( ζ, Z ) ≥ δ } . Remark 4.25 (feature of being ( δ, (cid:15) )-tame) . A key feature of an ( δ, (cid:15) ) -tame perturbed pseudoholomorphic map is that the f and (cid:15) always satisfy the hypotheses of Lemma 4.5. Lemma 4.26 (existence of tame perturbations) . Let ( M, η ) be a framed Hamiltonian manifold, let ( J, g ) be an η -adapted almost Her-mitian structure on the symplectization R × M , let ( u, S, j ) be a generally immersedpseudoholomorphic map, and let δ > satisfy δ < min (cid:16) dist γ (Crit a ◦ u , ∂S ) , min ζ ,ζ ∈Z ζ (cid:54) = ζ dist γ ( ζ , ζ ) , dist γ ( Z , ∂S ) (cid:17) . Then for each (cid:15) > satisfying (cid:15) < min(2 − , C B ) , where C B := sup (cid:8) (cid:107) B u ( ζ ) (cid:107) γ : dist γ ( ζ, Z ) ≥ δ (cid:9) , and B u is the second fundamental form of u , there exists a smooth map f : S → R for which (˜ u, ˜ , f, u, S, j ) is an ( δ, (cid:15) ) -tame perturbed pseudoholomorphic map in thesense of Definition 4.24.Proof. Let (cid:15) (cid:48) = (cid:15) (1+ C B ) , let h = a ◦ u , and apply Lemma A.8 from Section A.3. (cid:3) In order to proceed with later proofs, we would like to establish that for anygiven feral curve, outside some large compact set, the curve is usually immersed,and the tangent planes are usually close to being parallel to span( ∂ a , X η ). Our firstpass at making this precise is Lemma 4.27 below. Here the idea is that a tangentplane at a point ζ is close to being tangent to span( ∂ a , X η ) if and only if (cid:107) u ∗ λ ζ (cid:107) u ∗ g is nearly 1. Thus we are interested in the measure of the set of symplectization levelsets on which there are not many points with (cid:107) u ∗ λ (cid:107) u ∗ g < θ , for some specified value θ ∈ (0 , µ u ∗ g (cid:0) { ζ ∈ ( a ◦ u ) − ( t ) : (cid:107) ( u ∗ λ ) ζ (cid:107) u ∗ g < θ } (cid:1) should be smaller than some specified number δ >
0. Finally, in general the mea-sure of such level sets might of course be quite large, however Lemma 4.27 belowessentially states that it cannot be too large provided that the ω -energy is rathersmall; or more precisely, that for fixed δ and θ , the measure of such level sets isbounded in terms of the ω -energy. Thus for a feral curve, which has finite ω -energy,it should follow that outside a large compact set, the curve is usually immersed withtangent planes usually close to being parallel to span( ∂ a , X η ). This is now madeprecise with Lemma 4.27 below. Lemma 4.27 (tangent planes usually near vertical) . Let ( M, η ) be a framed Hamiltonian manifold, and ( J, g ) an η -adapted almost com-plex structure on R × M . Suppose further that ( u, S, j ) is a compact pseudoholo-morphic curve, possibly with boundary, with image in R × M , which satisfies thefollowing conditions.(1) (cid:82) S u ∗ ω ≤ E < ∞ (2) { ζ ∈ S : d ( a ◦ u )( ζ ) = 0 } ∩ ∂S = ∅ (3) u ( ∂S ) ⊂ { a , a } and a = sup { a ◦ u ( S ) } and a = inf { a ◦ u ( S ) } .With I := [ a , a ] , R u defined to be the regular values of a ◦ u , and for each θ ∈ (0 , and each δ > , we define the following set Q u,θ,δ := (cid:110) t ∈ R u : µ u ∗ g (cid:0) { ζ ∈ ( a ◦ u ) − ( t ) : (cid:107) ( u ∗ λ ) ζ (cid:107) u ∗ g < θ } (cid:1) > δ (cid:111) ERAL CURVES AND MINIMAL SETS 95
Then µ ( Q u,θ,δ ) ≤ E δ (1 − θ ) < ∞ . Here, µ is the Lebesgue measure associated to the coordinate a on R .Proof. For convenience, for each t ∈ R u and θ ∈ (0 ,
1) we will defineΓ t := { ζ ∈ S : a ◦ u ( ζ ) = t } and S θ := { ζ ∈ S : (cid:107) u ∗ λ ζ (cid:107) < θ } . Consequently, we may write Q u,θ,δ := (cid:8) t ∈ R u : µ u ∗ g (Γ t ∩ S θ ) > δ (cid:9) . Next we define the tangent vector fields ν and τ by ν := ∇ ( a ◦ u ) (cid:107)∇ ( a ◦ u ) (cid:107) γ and τ := jν where γ = u ∗ g , and g = da ⊗ da + λ ⊗ λ + ω ( · , J · ). It is straightforward to verifythe following properties, 0 = u ∗ λ ( ν ) = u ∗ da ( τ )0 < u ∗ da ( ν ) = u ∗ λ ( τ )1 = (cid:107) τ (cid:107) γ = (cid:107) ν (cid:107) γ , from which one can deduce that0 < da ( T u · ν ) = (cid:107)∇ ( a ◦ u ) (cid:107) u ∗ g ≤ , (cid:107) u ∗ λ (cid:107) u ∗ g = λ ( T u · τ )and 1 = ( λ ( T u · τ )) + ω ( T u · ν, T u · τ ) . From these we may estimate the measure of Q u,θ,δ as follows. µ ( Q u,θ,δ ) = (cid:90) Q u,θ,δ dt = (cid:90) { t ∈R u : µ u ∗ g (Γ t ∩ S θ ) >δ } dt = δ − (cid:90) { t ∈R u : µ u ∗ g (Γ t ∩ S θ ) >δ } δ dt ≤ δ − (cid:90) { t ∈R u : µ u ∗ g (Γ t ∩ S θ ) >δ } (cid:0) µ u ∗ g (Γ t ∩ S θ ) (cid:1) dt ≤ δ − (cid:90) I (cid:0) µ u ∗ g (Γ t ∩ S θ ) (cid:1) dt = δ − (cid:90) I (cid:90) Γ t ∩ S θ dµ u ∗ g dt = ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ (1 − θ ) dµ u ∗ g dt ≤ ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ − θ da ( T u · ν ) dµ u ∗ g dt ≤ ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ − (cid:107) u ∗ λ (cid:107) u ∗ g da ( T u · ν ) dµ u ∗ g dt = ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ − ( λ ( T u · τ )) da ( T u · ν ) dµ u ∗ g dt = ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ ω ( T u · ν, T u · τ ) da ( T u · ν ) dµ u ∗ g dt = ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ∩ S θ ω ( T u · ν, T u · τ ) (cid:107)∇ ( a ◦ u ) (cid:107) u ∗ g dµ u ∗ g dt ≤ ( δ (1 − θ )) − (cid:90) I (cid:90) Γ t ω ( T u · ν, T u · τ ) (cid:107)∇ ( a ◦ u ) (cid:107) u ∗ g dµ u ∗ g dt = ( δ (1 − θ )) − (cid:90) S ω ( T u · τ, T u · ν ) dµ u ∗ g = ( δ (1 − θ )) − (cid:90) S u ∗ ω ≤ E δ (1 − θ ) , where to achieve the second to last equality we have made use of Proposition 4.14with f = ω ( T u · ν, T u · τ ). (cid:3) In order to state Proposition 4.30 below concisely, it will be useful to have thefollowing definition at our disposal.
Definition 4.28 (connected component S ρ ( ζ )) . Let ( W, g ) be a Riemannian manifold with bounded geometry, , let S be manifold,and let u : S → W be a smooth map. For each ρ > and each ζ ∈ S , we define S ρ ( ζ ) to be the connected component of u − (cid:0) B ρ ( u ( ζ )) (cid:1) containing ζ ; here for each p ∈ W , the set B ρ ( p ) ⊂ W is the metric ball of radius ρ centered at p . Before proceeding, we need an additional geometric constant, namely r , theexistence of which is guaranteed by the following lemma. Lemma 4.29 (small radius r ) . Let ( M, g ) be a smooth Riemannian manifold of bounded geometry, and let λ ∈ Ω ( M ) be a smooth one-form with the property that for each point p ∈ M we have sup (cid:54) = τ ∈ T p M λ ( τ ) (cid:107) τ (cid:107) g = 1 Then there exists a positive real number r = r ( M, g, λ ) ≤ with the followingsignificance. For each smooth unit speed immersion ˜ q : [0 , T ] → M which satisfiesthe following conditions(1) λ (˜ q (cid:48) ( t )) > (2) r ≤ (cid:82) ˜ q λ ≤ r (3) µ q ∗ g ( { t ∈ [0 , T ] : λ (˜ q (cid:48) ( t )) < } ) ≤ r also satisfies dist g (cid:0) ˜ q (0) , ˜ q ( T ) (cid:1) ≥ r . Recall that a Riemannian manifold is said to have bounded geometry provided the sectionalcurvature is uniformly bounded from above and below and the injectivity radius of the manifoldis positive.
ERAL CURVES AND MINIMAL SETS 97
Proof.
We begin by letting ρ = min(1 , inj( M )) where inj( M ) is the injectivity radiusof M with respect to g . For each p ∈ M we let B ρ ( p ) denote the metric ball of radius ρ centered at p . Recall that for each point p ∈ M and each orthonormal frame for T p M one may define geodesic normal coordinates on B ρ ( p ) which are centered at p .We will denote such coordinates as x = ( x , . . . , x m ), in which case we can expressthe metric as g = (cid:80) mi,j =1 g ij ( x ) dx i ⊗ dx j , and our one-form as λ = (cid:80) mi =1 λ i ( x ) dx i .Note that for each p ∈ M there exists an orthonormal frame of T p M such thatthe associated geodesic normal coordinates have the property that λ i ( p ) = δ ,i dx ,where δ ,i is the Kronecker delta. We then fix r ∈ (0 , ρ ) sufficiently small sothat for any p ∈ M and any such orthonormal frame, we have(52) sup y ∈B r ( p ) (cid:107) dx y − λ y (cid:107) g ≤ . Next we consider an immersion ˜ q : [0 , T ] → M which satisfies the above hypothesesof the lemma. We now need to estimate the length of the path ˜ q in terms of r . Tothat end, we have:length(˜ q ) = T = (cid:90) T (cid:107) ˜ q (cid:48) ( t ) (cid:107) g dt = (cid:90) { t ∈ [0 ,T ]: λ (˜ q (cid:48) ( t )) < } (cid:107) ˜ q (cid:48) ( t ) (cid:107) g dt + (cid:90) { t ∈ [0 ,T ]:1 ≥ λ (˜ q (cid:48) ( t )) ≥ } (cid:107) ˜ q (cid:48) ( t ) (cid:107) g dt = µ q ∗ g (cid:0) { t ∈ [0 , T ] : λ (˜ q (cid:48) ( t )) < } (cid:1) + (cid:90) { t ∈ [0 ,T ]:1 ≥ λ (˜ q (cid:48) ( t )) ≥ } dt ≤ r + 2 (cid:90) { t ∈ [0 ,T ]:1 ≥ λ (˜ q (cid:48) ( t )) ≥ } λ (˜ q (cid:48) ( t )) dt ≤ r + 2 (cid:90) T λ (˜ q (cid:48) ( t )) dt ≤ r . As a consequence of this estimate, we see that the image of ˜ q is contained in themetric ball of radius 100 r centered at ˜ q (0), and hence inequality (52) holds alongthe image of ˜ q . As such, we take geodesic normal coordinates centered at p := ˜ q (0)as above so that λ i ( p ) = δ ,i dx , and we estimate as follows. x (cid:0) ˜ q ( T ) (cid:1) = (cid:90) T ddt (cid:0) x (˜ q ( t ) (cid:1) dt = (cid:90) T dx (˜ q (cid:48) ( t )) dt = (cid:90) T λ (˜ q (cid:48) ( t )) dt + (cid:90) T (cid:0) dx − λ (cid:1) (˜ q (cid:48) ( t )) dt ≥ (cid:90) ˜ q λ − T ≥ r − r ≥ r . Since dist g (cid:0) ˜ q ( T ) , ˜ q (0) (cid:1) = (cid:16) m (cid:88) i =1 (cid:0) x i (˜ q ( T )) (cid:1) (cid:17) ≥ x (˜ q ( T )) ≥ r , the desired result is immediate. (cid:3) The core proof.
Here we provide the complete proof of Theorem 5, whichessentially states that for each generally immersed feral pseudoholomorphic curve,there exists a large compact set in the symplectization with the property thatoutside this compact set, the curve has uniformly bounded connected-local area.That is, in a small ball the area of each connected component of the portion ofthe curve contained in the ball has universally bounded area. The first and mosttechnical step towards proving Theorem 5 is to prove Proposition 4.30, which is aspecial case. We accomplish this at present. Proposition 4.30 (connected-local area bound – special case) . Let ( M, η = ( λ, ω )) be a framed Hamiltonian manifold, and let ( J, g ) be an η -adapted almost Hermitian structure on the symplectization R × M . Let r ≤ be the positive constant associated to ( M, g, λ ) which is guaranteed by Lemma 4.29.Let r = 2 − min (cid:0) C − h , r (cid:1) , where C h is the ambient geometry constant establishedin Definition 4.11. For each generally immersed pseudoholomorphic map ( u, S, j ) satisfying the following conditions(LL1) S is homeomorphic to an annulus(LL2) a ◦ u ( ∂S ) = { a , a } with − min (cid:0) C − h , r (cid:1) ≤ a − a (LL3) (cid:8) ζ ∈ S : a ◦ u ( ζ ) ∈ { a , a } and d ( a ◦ u )( ζ ) = 0 (cid:9) = ∅ (LL4) sup ζ ∈ S a ◦ u ( ζ ) − inf ζ ∈ S a ◦ u ( ζ ) ≤ − min (cid:0) C − h , r (cid:1) (LL5) < (cid:82) S u ∗ ω ≤ r (cid:0) ( a − a ) − + 10 C h (cid:1) − (LL6) (cid:82) ( a ◦ u ) − ( a ) ∩ ∂S u ∗ λ ≥ r (LL7) µ u ∗ g (cid:0) { ζ ∈ ∂S : a ◦ u ( ζ ) = a and (cid:107) ( u ∗ λ ) ζ (cid:107) u ∗ g < } ) ≤ r ,also has the following property: For each ζ ∈ S with (cid:12)(cid:12) a ◦ u ( ζ ) − ( a + a ) (cid:12)(cid:12) ≤ ( a − a ) , we also have (53) Area u ∗ g (cid:0) S r ( ζ ) (cid:1) ≤ . Here, as above, µ u ∗ g is the one-dimensional Hausdorff measure associated to themetric u ∗ g . We note that the statement and proof of Proposition 4.30 are each rather long,so we take a moment to clarify the former and outline the latter. First, the hy-potheses require that we are dealing with a compact pseudoholomorphic curve,homeomorphic to an annulus, with a “top” boundary at the symplectization levelset { a } × M , and “bottom” boundary at the symplectization level set { a } × M .We allow that the interior points of the curve may lie either above { a } × M or Here by “small” we mean small relative to the geometry of the ambient manifold and notsmall relative to the curve itself.
ERAL CURVES AND MINIMAL SETS 99 below { a } × M , however we demand that both a and a be regular values of a ◦ u ,and if we define the ad hoc constant C := 2 − min( C − h , r )then we require 18 C ≤ a − a ≤ sup ζ ∈ S a ◦ u ( ζ ) − inf ζ ∈ S a ◦ u ( ζ ) ≤ C. Roughly then, both the height difference between the boundaries and the heightdifference between the absolute peak and absolute valley can neither be too large nortoo small. We also demand that the ω -energy be rather small, the λ -integral alongthe bottom boundary be rather large, and the measure of those points in the bottomboundary for which the tangent planes are not close to span( ∂ a , X η ) is rather small.After imposing all of these conditions, Proposition 4.30 then guarantees that thearea of a connected component of the portion of the curve that lives in a ball ofradius r which is centered near ( a + a ) is uniformly bounded; indeed, the boundis simply 1.Before outlining the proof, it is natural to ask how one is likely to find a curvewhich satisfies these conditions, so we sketch a candidate example. Indeed, considera feral curve which, for example, has an absolute minimum and no maximum, soit extends to { + ∞} × M . For simplicity, we assume that for some sufficientlylarge and generic a ∈ R , the set ( a ◦ u ) − (( a , ∞ ) × M ) is diffeomorphic to acylinder R × S . Note that this simplifying condition is essentially what makesProposition 4.30 only a special case of Theorem 5. Given such a curve, one thenconsiders values a and a which are very large, and which are regular values of a ◦ u . One then defines a compact annular curve by restricting the domain ofthe feral curve to ( a ◦ u ) − ([ a , a ] × M ), and then capping off excess boundarycomponents with appropriate disks. For this resulting curve, we see that bymaking a sufficiently large, condition (LL5) must be satisfied because feral curveshave finite ω -energy. Condition (LL6) follows essentially because if the λ -integralalong the bottom boundary did not get arbitrarily large, our feral curve would havefinite Hofer-energy; thus we assume this is not the case so we must be able to findmany large a for which condition (LL6) is satisfied. One then finds a for whichcondition (LL7) holds by a judicious application of Lemma 4.27.We now turn our attention to sketching the proof of Proposition 4.30. As a pre-liminary step, we give the overarching idea which motivates the proof. Namely, thekey conclusion is that the quantity Area u ∗ g (cid:0) S r ( ζ ) (cid:1) is bounded by some large uni-versal constant. That this constant is 1 instead of 10 (10 ) is essentially irrelevant.Also irrelevant to the main thrust of the argument is the fact that we have explic-itly specified r in terms of geometric constants, and we have bound the ω -energy (cid:82) S u ∗ ω in terms of geometric data. Instead, the key idea is to consider the case thatthere exists a sequence of such pseudoholomorphic annuli with the property thatas one progresses through the sequence, one can find a point ζ not near the bound-ary of the curve such that Area u ∗ g (cid:0) S r ( ζ ) (cid:1) → ∞ while r → (cid:82) S u ∗ ω → ω -energy, suppose that no matter how small one fixes a radius r , and no matter That such a capping procedure is possible is established later when needed. The subscripts denoting the index of the term in the sequence has been suppressed fornotational clarity.
00 J.W. FISH AND H. HOFER how large one fixes A ∈ R + , one can always find a point ζ ∈ S so that a ◦ u ( ζ ) ≥ A and the area of the connected component of u − ( B r ( u ( ζ ))), that contains ζ , is aslarge as we like while the ω -energy is as small as we like. One then aims to derive acontradiction by finding a region of S that contains S r ( ζ ) but which has boundedarea. Indeed, much of the proof is focused on finding this region of S which providesthe desired contradiction. That we can specify certain quantities, like r , the areabound, etc, in terms of geometric constants simply follows from taking some extracare with our estimates.Let us now turn our attention to describing that region in S that contains S r ( ζ ),but which has the desired area bound. As a first step, we impose some drasticsimplifying assumptions to get at the core argument. In particular, we begin byassuming that on our pseudoholomorphic annulus, there are no critical points ofthe function a ◦ u . We weaken this assumption in a moment, however in thissimplified case, we observe that every gradient trajectory of a ◦ u initiating in ∂ − S will terminate in ∂ +0 S . Geometrically then, all gradient flow lines extendfrom the bottom boundary to the top boundary of our annulus, without gettingtrapped at critical points. We then consider a compact interval I ⊂ ∂ − S whichhas small ˜ α -measure; that is, suppose (cid:82) I ˜ α = (cid:82) I u ∗ λ is small. Then consider thepseudoholomorphic strip u : Σ → R × M determined by I ; that is, with ∂ − Σ = I , ∂ +0 Σ ⊂ ∂ +0 S , and with the other portions of the boundary given as gradient flowlines. In this case, we haveArea u ∗ g (Σ) = (cid:90) Σ u ∗ da ∧ ˜ α + u ∗ ω = (cid:90) Σ d (cid:0) ( a ◦ u − a )˜ α (cid:1) − (cid:90) Σ ( a ◦ u − a ) d ˜ α + (cid:90) Σ u ∗ ω = ( a − a ) (cid:90) I ˜ α − (cid:90) Σ ( a ◦ u − a ) d ˜ α + (cid:90) Σ u ∗ ω ≤ ( a − a ) (cid:90) I ˜ α + ( a − a ) (cid:107) d ˜ α (cid:107) Area u ∗ g (Σ) + (cid:90) Σ u ∗ ω. At this point, invoke Lemma 4.12 which bounds (cid:107) d ˜ α (cid:107) in terms of the ambientgeometry constant, and note that a − a is small, so that we obtain an estimateof the form 12 Area u ∗ g (Σ) ≤ ( a − a ) (cid:90) I ˜ α + (cid:90) Σ u ∗ ω. Here we recall that ( a − a ) is small by our hypotheses, and so is (cid:82) Σ u ∗ ω , and (cid:82) I ˜ α issmall by assumption. Thus the area of Σ is bounded, and the region is determinedsimply by choosing an interval I ⊂ ∂ − S . The goal then becomes to show that S r ( ζ ) ⊂ Σ for some choice of I , which would essentially yield the desired bound.We almost do this. Instead, we partition ∂ − S into a bunch of small intervals sothat ζ is contained in exactly one of the corresponding strips Σ. We then showthat S r ( ζ ) cannot intersect both gradient-flow boundary portions of any strip Σassociated to our partition. This is achieved by a simple geodesic distance argumentcombined with Lemma 4.29. With this established, it then follows that S r ( ζ ) iscontained in the union of three consecutive strips, and fails to have non-trivialintersection with the outer-most gradient-flow boundary portions. Our previousarea estimate applies, but in triple, and this is sufficient to obtain the desired areabound on S r ( ζ ). ERAL CURVES AND MINIMAL SETS 101
Of course more generally, a ◦ u may indeed have critical points, so we nextconsider the case that a ◦ u is a Morse function. In this case, we note that gradienttrajectories that initiate at points in ∂ − S now terminate at either points in ∂ +0 S , orelse in critical points of a ◦ u of Morse index either 1 or 2. Note that there are onlyfinitely many points in ∂ − S with gradient flow lines that limit to critical points ofMorse index 1, but potentially a continuum which limit to critical points of Morseindex 2. Thus the goal is to show that the set of such points has small ˜ α measure.This follows essentially from Lemma 4.21, which guarantees that if the ˜ α -measureof such points were not small, then neither would the ω -energy, which in fact issmall. Knowing that in an ˜ α -measure theoretic sense, most points in ∂ − S havegradient flow lines that limit to points in ∂ +0 S , we can adapt our aforementionedargument to achieve the desired area bound.Next we note that a ◦ u need not be a Morse function, however if u is an immer-sion, then we may find a perturbed patch of pseudoholomorphic curve via a smallperturbing function f , so that for the resulting perturbed curve (˜ u, ˜ , f, u, S, j ), thefunction a ◦ ˜ u is indeed Morse, and the previous arguments essentially hold. In-deed, for f chosen suitably small enough, the ˜ u ∗ g -area bound on the patch yieldsthe desired u ∗ g -area bound on that same patch, which is the goal.Finally, we must worry about the case that u is not immersed. Unfortunately,our method to perturb the curve does not handle non-immersed points. Howeverthe assumptions of Theorem 5 guarantee that there are only finitely many interiornon-immersed points, and none on the boundary. Thus our goal will be to firstfind certain small neighborhoods of the non-immersed points Z , and perturb thecurve on the compliment of these neighborhoods so that in the larger region a ◦ ˜ u is Morse. We then show that the set of points in ∂ − S which are initial points ofgradient flow lines of the function a ◦ ˜ u which pass into the small neighborhood of Z has ˜ α -measure which is controlled by the ω -energy of the given pseudoholomorphicannulus, and hence essentially small. This procedure is similar to showing thatthe set of points limiting to local maxima has small ˜ α -measure, but a touch morecomplicated.It is perhaps worth mentioning that only at this point does it make considerablesense to have established estimates so explicitly in terms of geometric and universalconstants. The issue is that in order to establish Theorem 5, we must guarantee thatif the ω -energy of a curve is small and the ˜ α -measure of the bottom boundary ∂ − S is large, then, in an ˜ α -measure theoretic sense, most gradient trajectories startingin the bottom boundary ∂ − S end in the top boundary ∂ +0 S . Of course, this neednot be true if a ◦ u is not Morse, so we need to perturb our curve and we neededto show that our area and strip estimates hold for the perturbed curve; for exam-ple, this motivates Theorem 9, Lemma 4.21, and Lemma 4.22 to be established for perturbed curves, instead of simply pseudoholomorphic curves. Moreover, becauseour perturbation method does not extend across non-immersed points, there willbe small regions of unperturbed curve and we need to establish that most gradienttrajectories avoid these regions. A complication however is that as we make theneighborhood of the critical points smaller, our perturbing function f must alsochange, which in turn changes which gradient trajectories enter the neighborhood.Worse still, because the curvature of a curve may be unbounded in a neighborhoodof a non-immersed point, we see that the C norm of the function f must be made
02 J.W. FISH AND H. HOFER smaller as we shrink the neighborhood. Consequently, less exacting care in obtain-ing estimates can easily lead to circular reasoning: the size of the neighborhood ofthe non-immersed points depends on the gradient flow lines, which depend on theperturbing function f , which depends on the size of the neighborhood. To avoidthis circular logic, we are careful throughout this manuscript to make estimates andinequalities in terms of universal and geometric constants. Consequently, our choiceof neighborhood depends only on geometric constants associated to either the curveitself or the ambient geometry, and our perturbing function then depends upon theneighborhood, but not the other way around. The upshot is that we avoid circulardependence, but the downside is the seemingly pedantic focus on precision in theobtained inequalities.Although the above sketch accurately characterizes the proof of Theorem 5, theactual proof will be implemented in somewhat reverse order. Specifically, as follows:Step 1. Carefully define the neighborhoods of the non-immersed points, andthen define an appropriate perturbation of the curve.Step 2. Show that the ˜ α -measure of the initial points of gradient flow lines in ∂ − S which enter into the neighborhood of the non-immersed pointsis bounded in terms of an ambient geometry constant and the ω -energy.Step 3. Show that the ˜ α -measure of the initial points of gradient flow linesin ∂ − S which limit to local maxima of a ◦ ˜ u is bounded in terms ofan ambient geometry constant and the ω -energy.Step 4. Approximate the set of points in ∂ − S , that limit to points in ∂ +0 S ,from the inside by finitely many compact pairwise disjoint intervals.Step 5. Construct the desired partition, and associated patches of our curve.Step 6. Estimate the area of each of these patches, show that S r ( ζ ) is con-tained in the union of a consecutive triple of patches, and completethe proof.This completes the outline of the proof, so that finally we turn our attentiontoward the actual proof of Proposition 4.30. Proof of Proposition 4.30.
Step 1.
We begin by recalling a previously used notation. ∂ − S := ( ∂S ) ∩ ( a ◦ u ) − ( a ) ∂ +0 S := ( ∂S ) ∩ ( a ◦ u ) − ( a )Next, we let Z ⊂ S \ ∂S denote the set of non-immersed points of u . Recall that aconsequence of ( u, S, j ) being generally immersed is that Z is finite. It will also beconvenient to fix δ > δ < − min (cid:16) dist γ ( ∂S, Z ) , min z,z (cid:48) ∈Z z (cid:54) = z (cid:48) (cid:0) dist γ ( z, z (cid:48) ) (cid:1) , dist γ (cid:0) Crit a ◦ u , ∂S (cid:1)(cid:17) . Of course, “precision” is in the eye of the beholder, since many estimates can be substantiallysharpened. Indeed, we have made little effort to distinguish from 2 − etc, however suchadditional precision seems to have little utility in regards to our results. ERAL CURVES AND MINIMAL SETS 103
Here γ = u ∗ g . It is well known (see for example Lemma 2.9 of [12]) that for each z ∈ Z there exists a local holomorphic chart φ z : O ( z ) → O (0) ⊂ C (cid:39) R , andgeodesic normal coordinates Φ z : O ( u ( z )) → O (0) ⊂ C m (cid:39) R m , and 2 ≤ k z ∈ N ,such that φ z ( z ) = 0, Φ z ( u ( z )) = 0, andΦ z ◦ u ◦ φ − z ( w ) = ( w k z , , . . . ,
0) + F z ( w )where F z ( w ) = O ( | w | k z +1 ) and dF z ( w ) = O ( | w | k z ). For each z ∈ Z we may locallydefine the function r z : O ( z ) → R by the following: r z ◦ φ − z ( w ) = | w | k z , which issmooth everywhere it is defined, except possibly at w = 0 where it only must becontinuous (or more specifically, C , ). For each z ∈ Z we then define the sets V z := { ζ ∈ O ( z ) : r z ( ζ ) < δ } where we have assumed that δ > < δ ≤ δ < δ we have(1) { ζ ∈ V z : r z ( ζ ) = δ } ∼ = S (2) { ζ ∈ V z : r z ( ζ ) < δ } ∩ ∂S = ∅ (3) length γ (cid:0) { ζ ∈ V z : r z ( ζ ) = δ } (cid:1) ≤ πδk z (4) V z ∩ V z (cid:48) = ∅ for each z, z (cid:48) ∈ Z with z (cid:54) = z (cid:48) (5) δ ( (cid:80) z ∈Z k z ) ≤ C h π (cid:82) S u ∗ ω .For ease of notation, we now define V = (cid:91) z ∈Z V z and we define the function r : V → R by r (cid:12)(cid:12) V z = r z . We now fix δ > δ < min (cid:16) dist γ (Crit a ◦ u , ∂S ) , min z ,z ∈Z z (cid:54) = z dist γ ( z , z ) , δ (cid:17) , and(54) { ζ ∈ S : dist γ ( ζ, Z ) ≤ δ } ⊂ (cid:91) z ∈Z { ζ ∈ O ( z ) : r z ( ζ ) < δ } . We fix (cid:15) > (cid:15) < − min (cid:16)
11 + C B , C h , r (cid:17) where r is the small radius guaranteed by Lemma 4.29, C h is the ambient geometryconstant given in Definition 4.11, and C B := sup (cid:8) (cid:107) B u ( ζ ) (cid:107) γ : dist γ ( ζ, Z ) ≥ δ (cid:9) , and B u is the second fundamental form of u . We then let f : S → R be a smoothfunction for which (˜ u, ˜ , f, u, S, j ) is an ( δ, (cid:15) )-tame perturbed pseudoholomorphicmap in the sense of Definition 4.24; recall that the existence of such a perturbationis guaranteed by Lemma 4.26. Remark 4.31 (Morse failure) . By property (d3) of Definition 4.24, the function a ◦ ˜ u will fail to be Morse only in { ζ ∈ S : dist γ ( Z , ζ ) ≤ δ } , and equation (54) then guarantees that this function onlyfails to be Morse inside B = { ζ ∈ V : r ( ζ ) < δ } .
04 J.W. FISH AND H. HOFER
We will need to define several sets in terms of the following differential equation.(56) q : [0 , T ] → S q (cid:48) ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q ( s ) (cid:1) q (0) ∈ ∂ − S Here (cid:101) ∇ is the gradient with respect to (cid:101) γ = (cid:101) u ∗ g ; see also Definition 4.2. In particular,we define the sets A := (cid:110) ζ ∈ V : ∃ a solution to (56) s.t. q ( T ) = ζ (cid:111) (57) S := { ζ ∈ V : r ( ζ ) = δ }B := { ζ ∈ V : r ( ζ ) < δ }B (cid:48) := B ∩ AS (cid:48) := S ∩ A . Step 2.
By conditions on δ , we see that S ⊂ S is a finite set of pairwise disjoint embed-ded loops, which we equip with the subspace topology, and we let ψ : (cid:116) z ∈Z S → S denote a diffeomorphism. By Lemma 4.5, the following estimate holds.(58) length ˜ γ ( S ) ≤ · length γ ( S ) ≤ πδ (cid:88) z ∈Z k z ≤ C h (cid:90) S u ∗ ω. By existence, uniqueness, and continuous dependence upon initial conditions itfollows that A is open in S , and consequently S (cid:48) is open in S . Next we note that asa consequence of the definition of A , it follows that there is a well defined smoothmap π given by π : A → ∂ − Sπ ( ζ ) = ζ (cid:48) where there exists a solution to (56)for which q (0) = ζ (cid:48) and q ( T ) = ζ. By existence, uniqueness, and smooth dependence upon initial conditions, the map π is smooth. We define C ⊂ S (cid:48) to be the set of critical points of the restricted map π : S (cid:48) → ∂ − S , and we define S (cid:48)(cid:48) := S (cid:48) \ C . In other words, S (cid:48)(cid:48) consists of those points in S which are hit by gradient trajectoriesextending from points in ∂ − S but which are not critical points of π . Let cl S ( C )denote the closure in S of the set C , and observe that S (cid:48) ∩ cl S ( C ) = C , so that S (cid:48)(cid:48) is open in S . Next, we note that π ( C ) has measure zero, and hence ∂ − S \ π ( C ) isnon-empty, and thus we fix z (cid:48) ∈ ∂ − S \ π ( C ). Of course { z (cid:48) } is closed in ∂ − S , andhence π − ( z (cid:48) ) is closed in S (cid:48) , and thus S (cid:48) \ ( C ∪ π − ( z (cid:48) )) is open in S . As such, wedefine S (cid:48)(cid:48)(cid:48) := S (cid:48) \ (cid:0) C ∪ π − ( z (cid:48) ) (cid:1) = S (cid:48)(cid:48) \ π − ( z (cid:48) ) , which then must be open in S .Since S is diffeomorphic to the disjoint union of finitely many copies of S , and S (cid:48)(cid:48)(cid:48) is open in S , we conclude that S (cid:48)(cid:48)(cid:48) is diffeomorphic to the countable disjointunion of pairwise disjoint open intervals and copies of S . We immediately notehowever that it must in fact be the countable disjoint union of intervals with nocopies of S , essentially because we have removed π − ( z (cid:48) ) from S (cid:48)(cid:48) to obtain S (cid:48)(cid:48)(cid:48) , ERAL CURVES AND MINIMAL SETS 105 which forces each connected component of S (cid:48)(cid:48)(cid:48) to be diffeomorphic to an interval.Consequently, we write S (cid:48)(cid:48)(cid:48) = ∪ k ∈ M I k where each I k ⊂ S is diffeomorphic to an open interval, the I k are pairwise disjoint,the index set M denotes either a finite set or else N as appropriate, and π : I k → π ( I k ) ⊂ ∂ − S is a diffeomorphism for each k ∈ M . We note that by construction ofthe I k , the one-form ˜ α defines a one-dimensional volume form on each I k . However,for each k ∈ M , the map π : I k → π ( I k ) ⊂ ∂ − S is a diffeomorphism, and hence wemay define a second volume form π ∗ ˜ α on each I k . This creates a dichotomy: foreach k ∈ M , the orientations on I k induced from ˜ α and π ∗ ˜ α agree, or they do not.Thus we write {I k } k ∈ M = {I + k } k ∈ M + ∪ {I − k } k ∈ M − where {I + k } k ∈ M + denotes thoseintervals for which the orientations agree, and {I − k } k ∈ M − denotes those intervalsfor which the orientations disagree. Obviously then, S (cid:48)(cid:48)(cid:48) = (cid:16) (cid:91) k ∈ M + I + k (cid:17) (cid:91) (cid:16) (cid:91) k ∈ M − I − k (cid:17) . We now make the following claim.
Lemma 4.32 (a technical containment) . π ( C ) ∪ { z (cid:48) } ∪ π ( ∪ k ∈ M + I + k ) = π ( S (cid:48) ) . (59) Proof.
To prove this lemma, we first let ζ ∈ S (cid:48) , so there exists gradient trajectoryemanating from ζ ∈ ∂ − S and terminating at ζ ∈ S (cid:48) . We observe that there arefour possible cases. Case I. ζ = z (cid:48) . In this case, π ( ζ ) ∈ { z (cid:48) } . Case II. ζ (cid:54) = z (cid:48) and the vector (cid:101) ∇ ( a ◦ ˜ u ) is tangent to ∂ S at ζ . In this case ζ ∈ C ,and hence π ( ζ ) ∈ π ( C ). Case III. ζ (cid:54) = z (cid:48) and the vector (cid:101) ∇ ( a ◦ ˜ u ) is transverse to ∂ S at ζ and inwardpointing relative to B . It immediately follows that ζ ∈ ∪ k ∈ M − I − k , and hence π ( ζ ) ∈ π (cid:0) ∪ k ∈ M − I − k (cid:1) . Case IV. ζ (cid:54) = z (cid:48) and the vector (cid:101) ∇ ( a ◦ ˜ u ) is transverse to ∂ S at ζ and outwardpointing relative to B . It immediately follows that ζ ∈ ∪ k ∈ M + I + k , and hence π ( ζ ) ∈ π (cid:0) ∪ k ∈ M + I + k (cid:1) .We conclude that π ( S (cid:48) ) = π ( C ) ∪ { z (cid:48) } ∪ π ( ∪ k ∈ M + I + k ) ∪ π ( ∪ k ∈ M − I − k ) , and hence to establish equation (59), it is sufficient to prove that π (cid:0) ∪ k ∈ M − I − k (cid:1) \ π ( C ) ⊂ π (cid:0) ∪ k ∈ M + I + k (cid:1) . Note however, that if ζ ∈ ∪ k ∈ M − I − k \ π − ◦ π ( C ), then (cid:101) ∇ ( a ◦ ˜ u ) is pointing outwardrelative to B at ζ . Or in other words, following the gradient flow (cid:101) ∇ ( a ◦ ˜ u ) from ζ forsufficiently small but negative time, yields a point in B . However, cl( B ) ∩ ∂ − S = ∅ ,and hence we conclude that the gradient flow line initiating at ζ ∈ ∂ − S and
06 J.W. FISH AND H. HOFER terminating at ζ ⊂ ∪ k ∈ M − I − k \ π − ◦ π ( C ) must first intersect S (cid:48) in an inwardpointing direction, and this intersection must be transverse. It immediately followsthat π ( ζ ) ⊂ π ( ∪ k ∈ M + I + k ), and hence π (cid:0) ∪ k ∈ M − I − k (cid:1) \ π ( C ) ⊂ π (cid:0) ∪ k ∈ M + I + k (cid:1) , as required. This completes the proof of Lemma 4.32. (cid:3) Next, we choose n ∈ N sufficiently large so that (cid:12)(cid:12)(cid:12) (cid:90) π ( ∪ k ∈ M + I + k ) ˜ α − (cid:90) π ( ∪ n k =1 I + k ) ˜ α (cid:12)(cid:12)(cid:12) < C h (cid:90) S u ∗ ω (60)where the orientation on ∪ k I + k ⊂ ∂ − S is such that ˜ α is a volume form. Recall thatthe I + k are diffeomorphic to open intervals, and π : I + k → ∂ − S are diffeomorphismswith their images. As such, we may find sets {J k } n k =1 in S (cid:48) with the followingproperties:(1) each J k is diffeomorphic to a compact interval(2) π ( J k ) ∩ π ( J k (cid:48) ) = ∅ for k (cid:54) = k (cid:48) (3) for each k ∈ { , . . . , n } there exists k (cid:48) ∈ { , . . . , n } such that J k ⊂ I + k (cid:48) (4) the map π : J k → π ( J k ) ⊂ ∂ − S is a diffeomorphism(5) and finally, (cid:12)(cid:12)(cid:12) (cid:90) ∪ n k =1 π ( I + k ) ˜ α − (cid:90) ∪ n k =1 π ( J k ) ˜ α (cid:12)(cid:12)(cid:12) < C h (cid:90) S u ∗ ω. (61)As a consequence of the existence of such J k , we note that there exist perturbedpseudoholomorphic strips (˜ u k , (cid:101) S k , ˜ , f, u, S, j ) for which ∂ − (cid:101) S k = π ( J k ) and ∂ +0 (cid:101) S k = J k ⊂ (cid:91) k ∈ M + I + k ⊂ S . With these perturbed pseudoholomorphic strips established, we are now able toestimate as follows. (cid:12)(cid:12)(cid:12) (cid:90) π ( B (cid:48) ) ˜ α (cid:12)(cid:12)(cid:12) = (cid:90) π ( B (cid:48) ) ˜ α = (cid:90) π ( S (cid:48)(cid:48) ) ˜ α (See Lemma 4.33 below)= (cid:90) ∪ k ∈ M + π ( I + k ) ˜ α = (cid:16) (cid:90) ∪ k ∈ M + π ( I + k ) ˜ α − (cid:90) ∪ n k =1 π ( I + k ) ˜ α (cid:17) + (cid:90) ∪ n k =1 π ( I + k ) ˜ α ≤ C h (cid:90) S u ∗ ω + (cid:90) ∪ n k =1 π ( I + k ) ˜ α = C h (cid:90) S u ∗ ω + (cid:16) (cid:90) ∪ n k =1 π ( I + k ) ˜ α − (cid:90) ∪ n k =1 π ( J k ) ˜ α (cid:17) + (cid:90) ∪ n k =1 π ( J k ) ˜ α ≤ C h (cid:90) S u ∗ ω + (cid:90) ∪ n k =1 π ( J k ) ˜ α = C h (cid:90) S u ∗ ω + (cid:16) (cid:90) ∪ n k =1 π ( J k ) ˜ α − (cid:90) ∪ n k =1 J k ˜ α (cid:17) + 2 (cid:90) ∪ n k =1 J k ˜ α ERAL CURVES AND MINIMAL SETS 107 ≤ C h (cid:90) S u ∗ ω + 2 C h n (cid:88) k =1 (cid:90) (cid:101) S k ˜ u ∗ ω + 2 (cid:90) ∪ n k =1 J k ˜ α ≤ C h (cid:90) S ˜ u ∗ ω + 2 (cid:90) ∪ nk =1 J k ˜ α ≤ C h (cid:90) S ˜ u ∗ ω + 2 (cid:107) ˜ α (cid:107) L ∞ length ˜ γ ( ∪ nk =1 J k ) ≤ C h (cid:90) S ˜ u ∗ ω + 2 length ˜ γ ( ∪ nk =1 J k )(62) ≤ C h (cid:90) S ˜ u ∗ ω + 2 length ˜ γ ( S ) ≤ C h (cid:90) S ˜ u ∗ ω where to obtain the second equality we have made use of Lemma 4.33 below, toobtain the third equality we have made use of the fact that π ( C ) ∪ { z (cid:48) } has measurezero together with Lemma 4.32, to obtain the first inequality we have made use ofequation (60), to obtain the second inequality we have employed equation (61), toobtain the third inequality we have employed Lemma 4.21, to obtain the inequalityat (62) we have used (cid:107) ˜ α (cid:107) ˜ γ = (cid:107) − ˜ u ∗ da ◦ ˜ (cid:107) ˜ u ∗ g = (cid:107) ˜ u ∗ da (cid:107) ˜ u ∗ g ≤ (cid:107) da (cid:107) g = 1 , and to obtain the final inequality we have employed equation (58). Note that theabove inequality relies on the following equality. Lemma 4.33 (equality of ˜ α integrals) . (cid:90) π ( B (cid:48) ) ˜ α = (cid:90) π ( S (cid:48)(cid:48) ) ˜ α Proof.
Recall the definitions of B and B (cid:48) from (57). First observe that each con-nected component of B is homeomorphic to an open disk which is disjoint from ∂ − S , and ∂ B = S . Also observe that B (cid:48) ⊂ B and π : B (cid:48) → ∂ − S is well defined. Itfollows that for each ζ ∈ B (cid:48) there exists a ζ (cid:48) ∈ S (cid:48) such that π ( ζ ) = π ( ζ (cid:48) ). From thiswe conclude that π ( B (cid:48) ) ⊂ π ( S (cid:48) ) . Next observe that the definition of S (cid:48)(cid:48) guarantees that if ζ ∈ S (cid:48)(cid:48) , then the gradienttrajectory solving (56) intersects S (cid:48)(cid:48) transversely at ζ . It follows that π ( S (cid:48)(cid:48) ) ⊂ π ( B (cid:48) ), and hence we have π ( S (cid:48)(cid:48) ) ⊂ π ( B (cid:48) ) ⊂ π ( S (cid:48) ) . We then recall that S (cid:48)(cid:48) = S (cid:48) \ C where C is the set of critical points of the map π : S (cid:48) → ∂ − S , and hence by Sard’s theorem we conclude that π ( C ) has Lebesguemeasure zero, and thus we have π ( S (cid:48) ) \ π ( C ) ⊂ π ( S (cid:48)(cid:48) ) ⊂ π ( B (cid:48) ) ⊂ π ( S (cid:48) ) . Since π ( C ) has Lebesgue measure zero, it immediately follows that S (cid:48)(cid:48) and B (cid:48) differby a set of measure zero and hence (cid:90) π ( B (cid:48) ) ˜ α = (cid:90) π ( S (cid:48)(cid:48) ) ˜ α
08 J.W. FISH AND H. HOFER which is the desired result. This completes the proof of Lemma 4.33. (cid:3)
As a consequence, we have shown that(63) (cid:12)(cid:12)(cid:12) (cid:90) π ( B (cid:48) ) ˜ α (cid:12)(cid:12)(cid:12) ≤ C h (cid:90) S u ∗ ω. Thus we have shown that the ˜ α -measure of those points in ∂ − S with gradient flowlines that enter into our neighborhood of Z is bounded in terms of the ambientgeometry constant and ω -energy, the latter of which is assumed to be small. Step 3.
Next, we recall Remark 4.31, which observes that there there are only finitelymany critical points of a ◦ ˜ u in S \ B and each will be non-degenerate. As such, for k ∈ { , , } we define the finite sets M k := { ζ ∈ S \ B : d ( a ◦ ˜ u )( ζ ) = 0 and Index Morse ( ζ ) = k } . Note that M consists of local minima and therefore there cannot exist solutionsto the gradient equation(64) q : [0 , ∞ ) → S q (cid:48) ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q ( s ) (cid:1) q (0) ∈ ∂ − S which limit to a point in M . Also note that M consists of finitely many (non-degenerate) saddle-points and hence there are only finitely many solutions to (64)which limit to a point in M ; we denote the set of such initial conditions D . Itremains to consider those initial conditions in ∂ − S for which solutions to (64) limitto points in M . To that end, we fix (cid:15) (cid:48) > z ∈ M the set { ζ ∈ S : a ◦ ˜ u ( ζ ) = a ◦ ˜ u ( z ) − (cid:15) (cid:48) } contains a connected component, S z , contained in a small neighborhood of z in which there exist local coordinates,( s, t ), such that a ◦ ˜ u ( s, t ) = a ◦ ˜ u ( z ) − s − t , and furthermore (cid:15) (cid:48) has been chosensufficiently small so that (cid:88) z ∈M length ˜ γ ( S z ) ≤ C h (cid:90) S u ∗ ω. Observe that by construction, no trajectory initiating from ∂ − S may limit to apoint in M without transversally intersecting ∪ ζ ∈M S ζ . Defining E := (cid:110) ζ ∈ ∂ − S : ∃ a solution to (56) such that q (0) = ζ and q ( T ) ∈ ∪ z ∈M S z (cid:111) , we note that E is open, and hence we may find finitely many pair-wise disjointclosed intervals L k ⊂ ∂ − S with the property that (cid:12)(cid:12)(cid:12) (cid:90) E ˜ α (cid:12)(cid:12)(cid:12) = (cid:90) E ˜ α ≤ (cid:88) k (cid:90) L k ˜ α + C h (cid:90) S u ∗ ω. As above, for each L k one constructs a perturbed pseudoholomorphic strip, denoted(˜ u k , (cid:101) S k , ˜ k , f, u, S, j ), with the property that ∂ − (cid:101) S k = L k and ∂ +0 (cid:101) S k ⊂ ∪ z ∈M S z .Also as above, this yields a similar estimate: (cid:88) k (cid:90) L k ˜ α = (cid:88) k (cid:16) (cid:90) ∂ − (cid:101) S k ˜ α − (cid:90) ∂ +0 (cid:101) S k ˜ α (cid:17) + (cid:88) k (cid:90) ∂ +0 (cid:101) S k ˜ α ERAL CURVES AND MINIMAL SETS 109 ≤ C h (cid:88) k (cid:90) (cid:101) S k u ∗ ω + (cid:88) k (cid:90) ∂ +0 (cid:101) S k ˜ α ≤ C h (cid:90) S u ∗ ω + 2 (cid:88) z ∈M length ˜ γ ( S z ) ≤ C h (cid:90) S u ∗ ω. From this we conclude that(65) (cid:90) E ˜ α ≤ C h (cid:90) S u ∗ ω. Thus we have shown that the ˜ α -measure of those points in ∂ − S which havegradient flow lines that limit to local maxima of a ◦ ˜ u is bounded in terms of theambient geometry constant and the ω -energy, the latter of which is assumed to besmall. Step 4.
At this point we define the
T ⊂ ∂ − S to be the set of points for which thereexists a solution(66) q : [0 , T ] → S q (cid:48) ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q ( s ) (cid:1) for which(1) q (0) ∈ ∂ − S (2) q ( T ) ∈ ∂ +0 S (3) q ([0 , T ]) (cid:84) (cid:0) ∪ z ∈Z { ζ ∈ V z : r z ( ζ ) ≤ δ } (cid:1) = ∅ .We note that a consequence of Remark 4.31 and equation (54), the function a ◦ ˜ u is Morse on S \ ∪ z ∈Z { ζ ∈ V z : r z ( ζ ) ≤ δ } . As such, we conclude that T ⊂ ∂ − S is open in ∂ − S . We then claim the following inequalities are true: (cid:90) ( ∂ − S ) \T ˜ α ≤ (cid:90) π ( B (cid:48) ) ∪D∪E ˜ α ≤ C h (cid:90) S u ∗ ω. Observe that the first inequality follows from the fact that ( ∂ − S ) \ T ⊂ π ( B (cid:48) ) ∪D ∪ E , and the second inequality follows from the fact that D is finite together withinequalities (63) and (65). Since T is open in ∂ − S , we note that there exist finitelymany pairwise disjoint closed intervals {T k } Nk =1 , each contained in T , such that(67) N (cid:88) k =1 (cid:90) T k ˜ α ≥ (cid:90) ∂ − S ˜ α − C h (cid:90) S u ∗ ω. We pause for a moment to highlight the utility of these strips. Roughly speaking,the proof of Theorem 5 would be significantly simpler if every gradient trajectoryinitiating at a point in ∂ − S terminated at point in ∂ +0 S . Because this is not thecase, the next best scenario would be for there to exist a finite set of disjoint closedintervals in ∂ − S with the property that their ˜ α -measure was close to that of ∂ − S ,and with the property that every gradient trajectory starting in one of these inter-vals then terminated in ∂ +0 S . The intervals {T k } Nk =1 have precisely this property,and thus heuristically we should think of the associated perturbed pseudoholomor-phic strips as taking the place of S .
10 J.W. FISH AND H. HOFER
Step 5.
We now claim the following.
Lemma 4.34 (modest length gradient trajectories) . For each closed interval
I ⊂ ∂ − S satisfying (cid:90) I ˜ α ≥ (cid:0) ( a − a ) − + 10 C h (cid:1) (cid:90) S u ∗ ω there exists a solution to (68) q : [0 , T ] → S q (cid:48) ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q ( s ) (cid:1) such that q (0) ∈ I , q ( T ) ∈ ∂ +0 S , and (69) length ˜ γ (cid:0) q ([0 , T ]) (cid:1) ≤ ( a − a ) . Proof.
We begin by observing (cid:90) I∩ ( ∪ Nk =1 T k ) ˜ α = (cid:90) I ˜ α − (cid:90) I∩ ( ∂ − S \∪ Nk =1 T k ) ˜ α ≥ (cid:90) I ˜ α − (cid:90) ∂ − S \∪ Nk =1 T k ˜ α = (cid:90) I ˜ α − (cid:90) ∂ − S ˜ α + N (cid:88) k =1 (cid:90) T k ˜ α ≥ (cid:90) I ˜ α − C h (cid:90) S u ∗ ω ≥ a − a (cid:90) S u ∗ ω. However
I ∩ ( ∪ Nk =1 T k ) is a finite union of closed intervals, so that by Lemma 4.23and the inequality just established, it follows that there exists ζ ∈ I with theproperty that the gradient line extending from this point intersects ∂ +0 S in finitetime and it satisfies the length estimate (69). This completes the proof of Lemma4.34. (cid:3) To continue, it will be convenient to define the following. Given two points, ζ , ζ ∈ ∂ − S , we define I ζ ζ ⊂ ∂ − S to be the closed interval, oriented so that ˜ α isa volume form on I ζ ζ , and such that ∂ I ζ ζ = ζ − ζ . We now find a finite set ofpoints { ζ k } nk =1 ⊂ ∂ − S with the following properties. For each k ∈ { , . . . , n } wehave r < (cid:90) I ζk +1 ζk ˜ α < r , where ζ n +1 := ζ . We also require that if ζ (cid:96) / ∈ { ζ k , ζ k +1 } then ζ (cid:96) / ∈ I ζ k +1 ζ k . ByLemma 4.34, it follows that for each k ∈ { , . . . , n } there exists ζ − k ∈ I ζ k +1 ζ k withthe property that there exists a solution to(70) q k : [0 , T k ] → S q (cid:48) k ( s ) = (cid:101) ∇ ( a ◦ ˜ u ) (cid:0) q k ( s ) (cid:1) such that q k (0) = ζ − k , q k ( T k ) ∈ ∂ +0 S , and(71) length ˜ γ (cid:0) q k ([0 , T k ]) (cid:1) ≤ ( a − a ) . ERAL CURVES AND MINIMAL SETS 111
For each k ∈ { , . . . , n } we then define z − k := ζ − k . These points satisfy the propertythat for each k ∈ { , . . . , n } we have(72) r ≤ (cid:90) I z − k +1 z − k ˜ α ≤ r , where for notational convenience we have used z − n +1 = z − . Denoting z + k := q k ( T k ) ∈ ∂ +0 S , we now define Σ k ⊂ S to be the surface uniquely determinedby having boundary ∂ Σ k = q k ([0 , T k ]) (cid:91) q k +2 ([0 , T k +2 ]) (cid:91) I z − k +1 z − k (cid:91) I z + k +1 z + k . Here we have abused notation a bit to write I z + k +1 z + k ⊂ ∂ +0 S , though its meaningshould be clear form context. For later use, we make the following definition. ∂ − Σ k := q k ([0 , T k ]) and ∂ +1 Σ k := q k +2 ([0 , T k +2 ])(73)Observe that the { z − k } nk =1 satisfy the property that if z − (cid:96) / ∈ { z − k , z − k +1 } then z − (cid:96) / ∈I z − k +1 z − k , and hence the { Σ k } nk =1 have the property that if k (cid:54) = (cid:96) then Σ k ∩ Σ (cid:96) is eitherempty or consists of a single gradient trajectory. Step 6.
It will be convenient to estimate the area of Σ k which is done as follows. Area ˜ γ (Σ k ) ≤ (cid:90) Σ k ˜ u ∗ da ∧ ˜ α + (cid:90) Σ k ˜ u ∗ ω = (cid:90) Σ k d (cid:0) ( a ◦ ˜ u − a )˜ α (cid:1) − (cid:90) Σ k ( a ◦ ˜ u − a ) d ˜ α + (cid:90) Σ k ˜ u ∗ ω = ( a − a ) (cid:90) I z − k +1 z − k ˜ α − (cid:90) Σ k ( a ◦ ˜ u − a ) d ˜ α + (cid:90) Σ k ˜ u ∗ ω ≤ ( a − a )6 r + (cid:107) a ◦ ˜ u − a (cid:107) L ∞ (Σ k ) (cid:107) d ˜ α (cid:107) L ∞ (Σ k ) Area ˜ γ (Σ k ) + ≤ ( a − a )6 r + 116 C h C h ˜ γ (Σ k ) + The first inequality follows from Lemma 4.8, and the final inequality employsLemma 4.12 and property (LL4) from our assumptions in Proposition 4.30. Con-sequently, the desired area estimate is given asArea ˜ γ (Σ k ) ≤ ( a − a )24 r + ≤ . We are now prepared to finish the proof of Proposition 4.30. Indeed, let ζ ∈ S sothat (cid:12)(cid:12) a ◦ u ( ζ ) − ( a + a ) (cid:12)(cid:12) ≤ ( a − a ) , and define (cid:101) S r ( ζ ) to be the connected component of ˜ u − (cid:0) B r (˜ u ( ζ )) (cid:1) which contains ζ . Here B r ( p ) is the open metric ball of radius r centered at p ∈ R × M . Recallingthat r = 2 − min (cid:0) C − h , r (cid:1) ≤ − ( a − a ), we now claim the following. Lemma 4.35 ( ζ cannot be close to both sides simultaneously) . It cannot be the case that (cid:101) S r ( ζ ) ∩ ∂ − Σ k (cid:54) = ∅ and (cid:101) S r ( ζ ) ∩ ∂ +1 Σ k (cid:54) = ∅ for any k ∈ { , . . . , n } ; here the ∂ ± Σ k are defined in equation (73).
12 J.W. FISH AND H. HOFER
We will prove Lemma 4.35 momentarily, but for now we make use of it to com-plete the proof of Proposition 4.30. Indeed, as a consequence of Lemma 4.35 itmust be the case that if (cid:101) S r ( ζ ) ∩ ∂ Σ k (cid:54) = ∅ then (cid:101) S r ( ζ ) ⊂ Σ k − ∪ Σ k ∪ Σ k +1 .Consequently,(74) Area ˜ γ (cid:0) (cid:101) S r ( ζ ) (cid:1) ≤ . Next, we note that since ˜ u is an ( δ, (cid:15) )-tame perturbation of u with (cid:15) < r (recallinequality (55) and Remark 4.25) it follows that for each z ∈ S r ( ζ ) we have ˜ u ( z ) ∈B r ( u ( z )). Moreover since u ( S r ( ζ )) ⊂ B r ( u ( ζ )) it then follows that u ( S r ( ζ )) ⊂B r (˜ u ( ζ )). From this it follows that ˜ u ( S r ( ζ )) ⊂ B r (˜ u ( ζ )). In other words we haveshown that S r ( ζ ) ⊂ ˜ u − ( B r (˜ u ( ζ ))). Note that by definition S r ( ζ ) is connected,and hence contained in a connected component of ˜ u − ( B r (˜ u ( ζ ))). Also recall thatby definition, (cid:101) S r ( ζ ) is the connected component of ˜ u − ( B r (˜ u ( ζ ))) containing ζ .Consequently, to show that S r ( ζ ) ⊂ (cid:101) S r ( ζ ) it is sufficient to show that theyhave non-empty intersection, however this is obvious since they each contain ζ bydefinition. Thus we have shown S r ( ζ ) ⊂ (cid:101) S r ( ζ ) . Making use of this and equation (74), we have ≥ Area ˜ γ ( (cid:101) S r ( ζ )) ≥ Area ˜ γ ( S r ( ζ )) ≥ Area γ ( S r ( ζ )) , where the final inequality follows from combining Lemma 4.5 – particularly in-equality (15) – together with equation (137) from Section A.2 which expresses theHausdorff measure in terms of coordinates and a Riemannian metric. Since γ = u ∗ g ,the above estimate can be restated asArea γ ( S r ( ζ )) ≤ , which is also the desired inequality (53). Other than providing the proof of Lemma4.35, this completes the proof of Proposition 4.30. (cid:3) In order to complete the proof of Proposition 4.30 it only remains to proveLemma 4.35, which we do at present.
Proof of Lemma 4.35.
Because ∂ ± Σ k = q k +1 ± ([0 , T k +1 ± ]), and because of in-equality (71), it follows thatmax (cid:16) length ˜ γ ( ∂ − Σ k ) , length ˜ γ ( ∂ +1 Σ k ) (cid:17) ≤ ( a − a ) . In order to derive a contradiction, let us assume that (cid:101) S r ( ζ ) ∩ ∂ − Σ k (cid:54) = ∅ and (cid:101) S r ( ζ ) ∩ ∂ +1 Σ k (cid:54) = ∅ . Consequently, there exists a piece-wise smooth path β : [0 , → R × M such that β (0) = ˜ u ( z − k ) = u ( z − k ), β (1) = ˜ u ( z − k +1 ) = u ( z − k +1 ), and(75) length g ( β ([0 , ≤ ( a − a ) + 8 r . Indeed, this path is described by following ˜ u ( ∂ − Σ k ) from ˜ u ( z − k ) to some pointinside B r (˜ u ( ζ )), following the unique geodesic to ˜ u ( ζ ), following a geodesic in B r (˜ u ( ζ )) to a point in ˜ u ( ∂ +1 Σ k ), and then following ˜ u ( ∂ +1 Σ k ) to ˜ u ( z − k +1 ). In ERAL CURVES AND MINIMAL SETS 113 light of inequality (75), and the fact that r ≤ − ( a − a ), we conclude thatlength g ( β ([0 , ≤ ( a − a ), and hence(76) dist g (cid:0) ˜ u ( z k ) , ˜ u ( z k +1 ) (cid:1) ≤ ( a − a ) . We now parametrize ∂ − Σ k by φ : [0 , T ] → ∂ − Σ k so that (cid:107) φ (cid:48) (cid:107) ˜ u ∗ g = 1 and φ (0) = z − k and φ ( T ) = z − k +1 . Define ˜ q := ˜ u ◦ φ . Observe that ˜ q is a unit speed parametrizationof a path between ˜ u ( z − k ) and ˜ u ( z − k +1 ). We also claim the following hold.(1) λ (˜ q (cid:48) ( t )) > t ∈ [0 , T ],(2) r ≤ (cid:82) ˜ q λ ≤ r (3) µ q ∗ g ( { t ∈ [0 , T ] : λ (˜ q (cid:48) ( t )) < } ) ≤ r We take a moment to verify these properties. Recall that ∂ − Σ k ⊂ ∂S and this isa connected component of the preimage of a regular value of the function a ◦ u .Consequently we either have λ (˜ q (cid:48) ( t )) > t ∈ [0 , T ] or we have λ (˜ q (cid:48) ( t )) < t ∈ [0 , T ]; inequality (72) then establishes the former holds. The second propertyis simply a restatement of equation (72). The third property follows from combiningseveral observations, which we accomplish presently. Because ˜ u is an ( (cid:15), δ )-tameperturbation of u , it follows that ˜ u (cid:12)(cid:12) ∂S = u (cid:12)(cid:12) ∂S , and hence ˜ u (cid:12)(cid:12) ∂ − Σ k = u (cid:12)(cid:12) ∂ − Σ k . Because φ is a ˜ u ∗ g -unit speed parametrization, and because of the first property, and because˜ u ( ∂ − Σ k ) = a , it follows that pointwise we have (cid:107) u ∗ λ (cid:107) u ∗ g = λ (˜ q (cid:48) ). Combining thesefacts together with assumption ( LL
7) then yields r ≥ µ u ∗ g (cid:0) { ζ ∈ ∂S : a ◦ u ( ζ ) = a and (cid:107) u ∗ λ (cid:107) u ∗ g < } ) ≥ µ u ∗ g (cid:0) { ζ ∈ ∂ − Σ k : (cid:107) u ∗ λ (cid:107) u ∗ g < } )= µ u ∗ g (cid:0) { t ∈ [0 , T ] : λ (˜ q (cid:48) ( t )) < } ) , which is the claim of the third property. With these properties established, we nowapply Lemma 4.29, which guarantees that(77) dist g (cid:0) ˜ q (0) , ˜ q ( T ) (cid:1) ≥ r . However, r ≤ dist g (cid:0) ˜ q (0) , ˜ q ( T ) (cid:1) by (77)= dist g (cid:0) ˜ u ( z − k ) , ˜ u ( z − k +1 ) (cid:1) since ˜ u ( z − k ) = ˜ q (0) and ˜ u ( z − k +1 ) = ˜ q ( T ) ≤ ( a − a ) by (76) ≤ − r by ( LL r which is the desired contradiction. This completes the proof of Lemma 4.35. (cid:3) Lemma 4.36 (connected-local area bound for orbit cylinders) . Proposition 4.30 remains true when the assumption(LL5) < (cid:82) S u ∗ ω ≤ r (cid:0) ( a − a ) − + 10 C h (cid:1) − is weakened to the following(LL5’) (cid:82) S u ∗ ω ≤ r (cid:0) ( a − a ) − + 10 C h (cid:1) − .That is, we allow for the case that (cid:82) S u ∗ ω = 0 .
14 J.W. FISH AND H. HOFER
Proof.
We begin by observing that we need only prove the case that (cid:82) S u ∗ ω = 0.Since u : S → R × M is pseudoholomorphic and ω evaluates non-negatively on J -complex lines, it follows that in this case, there must exist a trajectory γ : R → M of the Hamiltonian vector field X η with the property that u ( S ) ⊂ R × γ ( R ).By property (LL2) we have a ◦ u ( ∂S ) = { a , a } , and by property (LL3) itfollows that there are no critical points of a ◦ u on ∂S . Consequently, u ∗ λ (cid:12)(cid:12) ∂S isnon-vanishing, and thus there must exist0 < T = (cid:90) ( a ◦ u ) − ( a ) ∩ ∂S u ∗ λ such that γ (0) = γ ( T ). Moreover, associated to the covering mapΦ : R × ( R /T Z ) → R × γ ( R )Φ( s, t ) = (cid:0) s, γ ( t ) (cid:1) there exists a lift φ : S → R × ( R /T Z ) of u : S → R × γ ( R ) ⊂ R × M . Equipping R × ( R /T Z ) with the almost complex structure J∂ s = ∂ t makes Φ pseudoholomorphicand hence φ is pseudoholomorphic. We then conclude from the maximum principlethat φ is an embedding of S into [ a , a ] × ( R /T Z ). Using φ to pull back coordinates( s, t ), we then have u ( s, t ) = ( s, γ ( t )). Consequently for each ζ ∈ S with (cid:12)(cid:12) a ◦ u ( ζ ) − ( a + a ) (cid:12)(cid:12) ≤ ( a − a )we also have Area u ∗ g (cid:0) S r ( ζ ) (cid:1) ≤ πr ≤ π (cid:16) − (cid:17) ≤ , which is the desired conclusion and completes the proof of Lemma 4.36. (cid:3) With the proof of Proposition 4.30 and Lemma 4.36 established, we are nowprepared to prove the main result of this section.
Theorem 5 (asymptotic connected-local area bound) . Let ( M, η ) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . Then the positive constant r = r ( M, η, J, g ) guaranteed by Proposition 4.30 has the following additional significance. For eachgenerally immersed feral pseudoholomorphic curve ( u, S, j ) in R × M , there existsa compact set of the form K := [ − a , a ] × M with the property that for each ζ ∈ S such that u ( ζ ) / ∈ K we have Area u ∗ g (cid:0) S r ( ζ ) (cid:1) ≤ here S r ( ζ ) is defined to be the connected component of u − ( B r ( u ( ζ ))) containing ζ , and B r ( p ) is the open metric ball of radius r centered at the point p ∈ R × M .Proof. Before we begin, we note that our proof will primarily rely on Proposition4.30 above to achieve the desired area bound. In a sense, or goal is to show thatoutside a large compact set, a point in a feral curve lives inside an annulus whichsatisfies the hypotheses of Proposition 4.30, and hence the desired area boundfollows immediately. As to be expected, the bulk of the work below is devoted toconstructing such an annulus.We begin by letting r denote the constant guaranteed by Proposition 4.30, andwe define R ± ⊂ R by R ± = { e ∈ R | e is regular for a ◦ u and − a ◦ u } . ERAL CURVES AND MINIMAL SETS 115
For notational convenience, we define (cid:15) := min( C − h , r ) , (78)where C h is the ambient geometry constant given in Definition 4.11, and r is thepositive constant provided in Lemma 4.29. We also let (cid:126) = (cid:126) ( M, η, J, g, − (cid:15) , C g )be the constant guaranteed by Theorem 4 with the genus bound C g := 0. Fornotational convenience, for each a ∈ R ± , we define S a = ( a ◦ u ) − (cid:0) ( − a , a ) (cid:1) and S a = ( a ◦ u ) − (cid:0) [ − a , a ] (cid:1) Observe that there exists an a ∈ R ± with the following properties.(L1) (cid:82) S \ S a u ∗ ω ≤ min (cid:0) , (cid:126) , − (cid:15) r , (2 (cid:15) − + 10 C h ) − (cid:1) (L2) Genus( S ) = Genus( S a )(L3) nc ( S \ S a ) = Punct( S )Recall the notion of Punct( S ) is given by Definition 1.3, and nc X denotes thenumber of path-connected components of X that are not compact. Note that forany a ∈ R ± for which a > a , properties (L1) - (L3) hold even when S a isreplaced with S a .Recall that our goal here is to show that for some sufficiently large c >
0, theportion of the pseudoholomorphic curve in the complement of [ − c, c ] × M satisfiesthe aforementioned connected-local area bound. Strictly speaking, this breaks ourproblem up into two cases: the portion of curve in ( c, ∞ ) × M and the portion in( −∞ , − c ) × M , however we shall henceforth only study the first case; the second isessentially identical.Suppose a (cid:48) , b (cid:48) ∈ R ± so that a + (cid:15) < a (cid:48) < b (cid:48) and consider S b (cid:48) a (cid:48) , where here andthroughout we use the notation S yx := ( a ◦ u ) − (( x, y )) and S yx := ( a ◦ u ) − ([ x, y ]) . We characterize ∂S b (cid:48) a (cid:48) by separating it into essential and inessential components.More specifically we write ∂S b (cid:48) a (cid:48) = ∂ ess S b (cid:48) a (cid:48) ∪ ∂ ⊥ ess S b (cid:48) a (cid:48) where ∂ ess S b (cid:48) a (cid:48) consists of thoseconnected components of ∂S b (cid:48) a (cid:48) which are contained in either non-compact connectedcomponents of S \ S b (cid:48) a (cid:48) , or are contained in connected components of S \ ( S b (cid:48) a (cid:48) ∪ S a − a )which have non-trivial intersection with both S a − a and S b (cid:48) a (cid:48) . More geometrically, ifwe think of S a − a as being the core of S , then the essential boundary componentsof S b (cid:48) a (cid:48) are those which connect S b (cid:48) a (cid:48) to the infinite positive end, or else they areboundary components of portions of curves which connect S b (cid:48) a (cid:48) to the core of S .Observe that by definition of essential boundary components, we may cap offthe inessential boundary components with the union of connected components of S \ ( S a −∞ ∪ S b (cid:48) a (cid:48) ) which satisfy the following conditions(1) the connected component is compact(2) the connected component has empty intersection with S a − a We will let (cid:101) S b (cid:48) a (cid:48) denote the union of S b (cid:48) a (cid:48) with the union of these specified cappingcomponents, so that ∂ (cid:101) S b (cid:48) a (cid:48) consists only of essential components. Summarizing, wehave constructed (cid:101) S b (cid:48) a (cid:48) so that( (cid:101) S (cid:101) S b (cid:48) a (cid:48) is compact with a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ) ⊂ { a (cid:48) , b (cid:48) }
16 J.W. FISH AND H. HOFER ( (cid:101) S u ( (cid:101) S b (cid:48) a (cid:48) ) ⊂ ( a , ∞ ) × M ( (cid:101) S ∂ (cid:101) S b (cid:48) a (cid:48) is contained in the union of connected components of S \ ( S a −∞ ∪ S b (cid:48) a (cid:48) )which are either non-compact, or have non-empty intersection with both S a −∞ and S b (cid:48) a (cid:48) .We now claim the following. Lemma 4.37 (short capping disks) . max (cid:0) a (cid:48) − inf ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) , sup ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) − b (cid:48) (cid:1) ≤ − (cid:15) . Proof.
Suppose not. For example, suppose(79) a (cid:48) − inf ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) > − (cid:15) . Then there must exist a non-empty connected component (cid:98) S of S \ ( S a −∞ ∪ S ∞ a (cid:48) )with the following properties:(1) ( u, (cid:98) S, j ) is compact, connected, and generally immersed(2) a ◦ u ( S ) ⊂ ( a , ∞ )(3) Genus( (cid:98) S ) = 0(4) a ◦ u ( ∂ (cid:98) S ) = { a min + c (cid:48) } ,where a min := inf ζ ∈ (cid:98) S a ◦ u ( ζ ), and c (cid:48) > − (cid:15) . We now apply Theorem 4 with C g = 0 and r = 2 − (cid:15) to conclude(80) (cid:90) (cid:98) S u ∗ ω ≥ (cid:126) , where (cid:126) = (cid:126) ( M, η, J, g, − (cid:15) , (cid:98) S ⊂ S and a ◦ u ( (cid:98) S ) ⊂ ( a , ∞ ) contradicts the fact that a has been chosen sothat 0 < (cid:126) ≤ (cid:90) (cid:98) S u ∗ ω ≤ (cid:90) S \ S a − a u ∗ ω ≤ (cid:126) . This shows that inequality (79) is impossible. A similar argument shows that wemust also have sup ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) − b (cid:48) ≤ − (cid:15) . This completes the proof of Lemma 4.37 (cid:3)
In light of Lemma 4.37, we conclude that another property of (cid:101) S b (cid:48) a (cid:48) is the following.( (cid:101) S
4) sup ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) − inf ζ ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ ) ≤ b (cid:48) − a (cid:48) + 2 − (cid:15) .It will be useful to employ the following notation: If X is a topological space,then we let Comp( X ) denote the set of connected components of X . It will alsobe useful to say that connected components L , L ∈ Comp( ∂S a − a ) are eventuallyconnected if there exists a connected component ˇ S of S \ S a − a for which L ∪ L ⊂ ˇ S .We now note that because u : S → R × M is a proper map and a is a regular valueof both a ◦ u and − a ◦ u , it follows that Comp( ∂S a − a ) is finite. Consequently, thereexists a ∈ R + for which a > a and with the following property. For each pair L , L ∈ Comp( ∂S a − a ) which are eventually connected, there exists a connectedcomponent ˇ S of S a − a \ S a − a for which L ∪ L ⊂ ˇ S . ERAL CURVES AND MINIMAL SETS 117
We henceforth consider (cid:101) S b (cid:48) a (cid:48) with a (cid:48) > a . A consequence of this assumption isthat each connected component of (cid:101) S b (cid:48) a (cid:48) only has at most one bottom boundary com-ponent, and at most one top component. We make this precise with the followingtwo lemmas. Lemma 4.38 (bottom boundary is a circle or empty) . Consider (cid:101) S b (cid:48) a (cid:48) with a (cid:48) , b (cid:48) ∈ R ± and for which a (cid:48) > a . Then each connectedcomponent ˇ S of (cid:101) S b (cid:48) a (cid:48) has the property that Comp (cid:0) ∂ ˇ S ∩ ( a ◦ u ) − ( a (cid:48) ) (cid:1) ≤ . Proof.
Suppose not; that is, suppose there exists a connected component ˇ S of (cid:101) S b (cid:48) a (cid:48) for which Comp (cid:0) ∂ ˇ S ∩ ( a ◦ u ) − ( a (cid:48) ) (cid:1) ≥ . Then by definition of a and the fact that a (cid:48) > a , there exists c (cid:48) ∈ R ± forwhich a < c (cid:48) < a (cid:48) and has the property that there exists a connected componentˇ S of S a (cid:48) c (cid:48) for which ∂ ˇ S ∩ ( a ◦ u ) − ( a (cid:48) ) ⊂ ˇ S . We now define n +1 = (cid:0) ˇ S ∩ ( a ◦ u ) − ( b (cid:48) ) (cid:1) ≥ n − = (cid:0) ˇ S ∩ ( a ◦ u ) − ( a (cid:48) ) (cid:1) ≥ n = (cid:0) ∂ ˇ S (cid:1) − n − ≥ , where X denotes the number of connected components of X . Recall that for acompact two-dimensional surface S possibly with boundary, the Euler characteristicof S is given by χ ( S ) = 2 − S ) − ∂S ) , and thus χ ( ˇ S ) = 2 − − ( n +1 + n − ) χ ( ˇ S ) = 2 − − ( n + n − ) . We now make two observations; first ˇ S ∩ ˇ S = ∂ ˇ S ∩ ( a ◦ u ) − ( a (cid:48) ). Second,ˇ S ∪ ˇ S ⊂ S is a compact two dimensional surface with boundary, and whichsatisfies ∂ ( ˇ S ∪ ˇ S )) = n +1 + n . We then compute χ ( ˇ S ∪ ˇ S ) = χ ( ˇ S ) + χ ( ˇ S )= (cid:0) − − ( n +1 + n − ) (cid:1) + (cid:0) − − ( n + n − ) (cid:1) = 2 − n − − − ( n +1 + n )= 2 − n − − − ∂ ( ˇ S ∪ ˇ S ))= 2 − S ∪ ˇ S ) − ∂ ( ˇ S ∪ ˇ S )) , and conclude that Genus( ˇ S ∪ ˇ S ) = n − − >
0. However, from this it immediatelyfollows that Genus( S a − a ) < Genus( S ) which is impossible by the definition of a and genus super-additivity, namely Lemma 2.29. This is the desired contradictionwhich proves Lemma 4.38. (cid:3)
18 J.W. FISH AND H. HOFER
Lemma 4.39 (top boundary is a circle or empty) . Consider (cid:101) S b (cid:48) a (cid:48) with a (cid:48) , b (cid:48) ∈ R ± and for which a (cid:48) > a . Then each connectedcomponent ˇ S of (cid:101) S b (cid:48) a (cid:48) has the property that Comp (cid:0) ∂ ˇ S ∩ ( a ◦ u ) − ( b (cid:48) ) (cid:1) ≤ . Proof.
Suppose not. Then by Definition 1.3 and Remark 1.4 we have Punct( S b (cid:48) − b (cid:48) ) > Punct( S a (cid:48) − a (cid:48) ) which contradicts the fact that Punct( S a − a ) = Punct( S ) and the factthat S b b ⊂ S b (cid:48) b (cid:48) implies Punct( S b b ) ≤ Punct( S b (cid:48) b (cid:48) ) . This contradiction completes the proof. (cid:3)
With the above topological preliminaries out of the way, we now turn our at-tention to more measure theoretic preliminaries. Indeed, for the remainder of theproof fix ζ ∈ S such that a ◦ u ( ζ ) ≥ a + 2. We also note that to prove Theorem 5we must establish the existence of a compact set K ⊂ R × M , which we can nowdefine explicitly as K := [ − a − , a + 2] × M. Next, we consider surfaces S b (cid:48)(cid:48) + b (cid:48)(cid:48)− and S a (cid:48)(cid:48) + a (cid:48)(cid:48)− where a (cid:48)(cid:48)− , a (cid:48)(cid:48) + , b (cid:48)(cid:48)− , b (cid:48)(cid:48) + ∈ R ± and a ◦ u ( ζ ) + 2 − (cid:15) − − (cid:15) ≤ b (cid:48)(cid:48) + ≤ a ◦ u ( ζ ) + 2 − (cid:15) a ◦ u ( ζ ) + 2 − (cid:15) ≤ b (cid:48)(cid:48)− ≤ a ◦ u ( ζ ) + 2 − (cid:15) + 2 − (cid:15) a ◦ u ( ζ ) − − (cid:15) − − (cid:15) ≤ a (cid:48)(cid:48) + ≤ a ◦ u ( ζ ) − − (cid:15) a ◦ u ( ζ ) − − (cid:15) ≤ a (cid:48)(cid:48)− ≤ a ◦ u ( ζ ) − − (cid:15) + 2 − (cid:15) . Observe that by definition, we have b (cid:48)(cid:48) + − b (cid:48)(cid:48)− ≥ − (cid:15) − − (cid:15) − − (cid:15) − − (cid:15) ≥ − (cid:15) and similarly(81) a (cid:48)(cid:48) + − a (cid:48)(cid:48)− ≥ − (cid:15) . Likewise, it is elementary to establish2 − (cid:15) ≤ b (cid:48)(cid:48)− − a (cid:48)(cid:48) + and b (cid:48)(cid:48) + − a (cid:48)(cid:48)− ≤ − (cid:15) . (82)It is perhaps worth explicitly observing that a (cid:48)(cid:48)− < a (cid:48)(cid:48) + < a ◦ u ( ζ ) < b (cid:48)(cid:48)− < b (cid:48)(cid:48) + . We define Q u, ,r ( S b (cid:48)(cid:48) + b (cid:48)(cid:48)− ) := (cid:110) t ∈ [ b (cid:48)(cid:48)− , b (cid:48)(cid:48) + ] ∩R ± : µ u ∗ g (cid:0) { ζ ∈ ( a ◦ u ) − ( t ) : (cid:107) ( u ∗ λ ) ζ (cid:107) u ∗ g < } (cid:1) > r (cid:111) and Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ) := (cid:110) t ∈ [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ] ∩R ± : µ u ∗ g (cid:0) { ζ ∈ ( a ◦ u ) − ( t ) : (cid:107) ( u ∗ λ ) ζ (cid:107) u ∗ g < } (cid:1) > r (cid:111) . By construction we have Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ) ⊂ [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ] and Q u, ,r ( S b (cid:48)(cid:48) + b (cid:48)(cid:48)− ) ⊂ [ b (cid:48)(cid:48)− , b (cid:48)(cid:48) + ].However, as a consequence of Lemma 4.27, we also have µ (cid:0) Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ) (cid:1) ≤ r (1 − ( ) ) (cid:90) S a (cid:48)(cid:48) + a (cid:48)(cid:48)− u ∗ ω ERAL CURVES AND MINIMAL SETS 119 = 43 r (cid:90) S a (cid:48)(cid:48) + a (cid:48)(cid:48)− u ∗ ω ≤ r (cid:90) S \ S a − a u ∗ ω ≤ − (cid:15) , where to obtain final inequality we have made use of property ( L Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ) ⊂ [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ]satisfies µ (cid:0) Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ) (cid:1) ≤ − (cid:15) < − (cid:15) ≤ µ (cid:0) [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ] (cid:1) , where we have made use of inequality (81). We conclude that there exists a (cid:48) ∈ R ± ∩ [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ] \ Q u, ,r ( S a (cid:48)(cid:48) + a (cid:48)(cid:48)− ), and similarly there exists b (cid:48) ∈ R ± ∩ [ b (cid:48)(cid:48)− , b (cid:48)(cid:48) + ] \ Q u, ,r ( S b (cid:48)(cid:48) + b (cid:48)(cid:48)− ).In other words, there exists a (cid:48) ∈ [ a (cid:48)(cid:48)− , a (cid:48)(cid:48) + ] and b (cid:48) ∈ [ b (cid:48)(cid:48)− , b (cid:48)(cid:48) + ] which are each regularvalues of a ◦ u , and(83) µ u ∗ g (cid:0) { ζ (cid:48) ∈ ( a ◦ u ) − ( a (cid:48) ) : (cid:107) u ∗ λ (cid:107) u ∗ g < } (cid:1) ≤ r and(84) µ u ∗ g (cid:0) { ζ (cid:48) ∈ ( a ◦ u ) − ( b (cid:48) ) : (cid:107) u ∗ λ (cid:107) u ∗ g < } (cid:1) ≤ r . Importantly, we henceforth assume that a (cid:48) , b (cid:48) ∈ R ± have been fixed so that equation(83) and equation (84) are true, and so that a (cid:48)(cid:48)− ≤ a (cid:48) ≤ a (cid:48)(cid:48) + ≤ a ◦ u ( ζ ) ≤ b (cid:48)(cid:48)− ≤ b (cid:48) ≤ b (cid:48)(cid:48) + . It then follows from equation (82) that2 − (cid:15) ≤ b (cid:48)(cid:48)− − a (cid:48)(cid:48) + ≤ b (cid:48) − a (cid:48) ≤ b (cid:48)(cid:48) + − a (cid:48)(cid:48)− ≤ − (cid:15) , or for clarity, 2 − (cid:15) ≤ b (cid:48) − a (cid:48) ≤ − (cid:15) . (85)With these measure theoretic preliminaries out of the way, we can now com-plete the proof of Theorem 5. Consider the surface S b (cid:48) a (cid:48) , and more importantly, itsextension (cid:101) S b (cid:48) a (cid:48) . Furthermore, we let (cid:101) S b (cid:48) a (cid:48) ( ζ ) denote the connected component of (cid:101) S b (cid:48) a (cid:48) containing ζ . We list some properties of (cid:101) S b (cid:48) a (cid:48) ( ζ ) which have already been established.The properties (T1)–(T7) are listed in such a way that they can be compared withthe hypotheses (LL1)–(LL7) of Proposition 4.30. We note that (T6) is still emptyand will be filled during the discussion.(T1) (cid:101) S b (cid:48) a (cid:48) ( ζ ) is homeomorphic to either a sphere, a disk, or an annulus; this followsfrom Lemma 4.38 and Lemma 4.39, which guarantee that (cid:101) S b (cid:48) a (cid:48) ( ζ ) has at mosttwo boundary components, together with the fact that (cid:101) S b (cid:48) a (cid:48) ( ζ ) ⊂ S \ S a − a ,which, by definition of a and Lemma 2.29, guarantees Genus( (cid:101) S b (cid:48) a (cid:48) ( ζ )) = 0.(T2) a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ )) ⊂ { a (cid:48) , b (cid:48) } with 2 − (cid:15) ≤ b (cid:48) − a (cid:48) ; this follows from equation (85).(T3) { ζ (cid:48) ∈ (cid:101) S b (cid:48) a (cid:48) ( ζ ) : a ◦ u ( ζ (cid:48) ) ∈ { a (cid:48) , b (cid:48) } and d ( a ◦ u )( ζ (cid:48) ) = 0 } = ∅ ; this follows since a (cid:48) and b (cid:48) are regular values of a ◦ u .
20 J.W. FISH AND H. HOFER (T4) sup ζ (cid:48) ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ (cid:48) ) − inf ζ (cid:48) ∈ (cid:101) S b (cid:48) a (cid:48) a ◦ u ( ζ (cid:48) ) ≤ − (cid:15) ; this follows from property( (cid:101) S
4) combined with inequality (85):sup − inf ≤ b (cid:48) − a (cid:48) + 2 − (cid:15) ≤ − (cid:15) + 2 − (cid:15) ≤ − (cid:15) . which is the desired inequality.(T5) (cid:82) (cid:101) S b (cid:48) a (cid:48) u ∗ ω ≤ r (cid:0) ( b (cid:48) − a (cid:48) ) − + 10 C h ) − ; this follows from the fact that (cid:101) S b (cid:48) a (cid:48) ( ζ ) ⊂ S \ S a − a , the definition of a , property (L1), and the fact that b (cid:48) − a (cid:48) ≥ − (cid:15) (T6) — See the following discussion.(T7) µ u ∗ g (cid:0) { ζ (cid:48) ∈ ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ ) : a ◦ u ( ζ (cid:48) ) = a (cid:48) and (cid:107) u ∗ λ (cid:107) u ∗ g < } (cid:1) ≤ r ; this followsfrom our definition of a (cid:48) , and specifically equation (83).We also claim that(86) (cid:12)(cid:12) a ◦ u ( ζ ) − ( b (cid:48) + a (cid:48) ) (cid:12)(cid:12) ≤ ( b (cid:48) − a (cid:48) ) . Before justifying inequality (86) it may be helpful to recall that we have defined (cid:15) = min( C − h , r ). We note that from properties (T1) - (T7) and equation (86), toapply Proposition 4.30 (or Lemma 4.36 in the case that (cid:82) (cid:101) S b (cid:48) a (cid:48) u ∗ ω = 0), it is sufficientto establish that (cid:101) S b (cid:48) a (cid:48) ( ζ ) is an annulus with image of one boundary component in { a (cid:48) } × M and the other in { b (cid:48) } × M , where the condition (T6) given by (cid:90) ( a ◦ u ) − ( a (cid:48) ) ∩ ∂ (cid:101) S b (cid:48) a (cid:48) u ∗ λ ≥ r . We will do this momentarily, however at present we establish inequality (86). Tothat end, we begin by letting b (cid:48) = a ◦ u ( ζ ) + 2 − (cid:15) + (cid:15) b with 0 ≤ (cid:15) b ≤ − (cid:15) a (cid:48) = a ◦ u ( ζ ) − − (cid:15) − (cid:15) a with 0 ≤ (cid:15) a ≤ − (cid:15) and we define (cid:15) c := max( (cid:15) b , (cid:15) a ). We then observe that | (cid:15) b − (cid:15) a | ≤ (cid:15) c ≤ − (cid:15) ≤ − (cid:15) + ( (cid:15) b + (cid:15) a ) . so that(87) | (cid:15) b − (cid:15) a | ≤ (cid:0) − (cid:15) + ( (cid:15) b + (cid:15) a ) (cid:1) . However,(88) (cid:12)(cid:12) a ◦ u ( ζ ) − ( b (cid:48) + a (cid:48) ) (cid:12)(cid:12) = | (cid:15) b − (cid:15) a | and(89) ( b (cid:48) − a (cid:48) ) = (2 − (cid:15) + (cid:15) b + (cid:15) a ) . Combining equations (87) - (89) then establishes inequality (86). To complete theproof of Theorem 5, we break the problem into cases.
Case I: a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ )) (cid:54) = { a (cid:48) , b (cid:48) } . In this case we will assume that a (cid:48) / ∈ a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ )); the case that b (cid:48) / ∈ a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ )) follows in identical fashion. Next wedefine the surface ˇ S := (cid:101) S b (cid:48) a (cid:48) ∩ ( a ◦ u ) − [ a (cid:48) − − (cid:15) , b (cid:48) ]. We note that as a consequence ERAL CURVES AND MINIMAL SETS 121 of Lemma 4.37, ˇ S is indeed a smooth surface, possibly with smooth boundary, and a ◦ u ( ∂ ˇ S ) ⊂ { b (cid:48) } . Furthermore,sup ζ (cid:48) ∈ ˇ S a ◦ u ( ζ (cid:48) ) − inf ζ (cid:48) ∈ ˇ S a ◦ u ( ζ (cid:48) ) ≤ b (cid:48) − a (cid:48) + 2 − (cid:15) ≤ − (cid:15) . As a consequence of Theorem 9, we then have(90) Area u ∗ g ( ˇ S ) = (cid:90) ˇ S u ∗ ( da ∧ λ + ω ) ≤ e C h − (cid:15) (cid:90) ˇ S u ∗ ω ≤ e C h − (cid:15) ≤ . Recall that r = 2 − (cid:15) , and S r ( ζ ) is defined to be the connected component of u − ( B r ( u ( ζ ))) containing ζ , where B r ( p ) is a metric ball in R × M of radius r centered at the point p . Writing a † = a ◦ u ( ζ ), we then have S r ( ζ ) ⊂ S a † +2 − (cid:15) a † − − (cid:15) ( ζ ) ⊂ S a † +2 − (cid:15) a † − − (cid:15) ( ζ ) ⊂ S b (cid:48) a (cid:48) ( ζ ) ⊂ ˇ S, where we have let S c c ( ζ ) denote the connected component of S c c containing ζ .Combining this containment with (90) then yieldsArea u ∗ g ( S r ( ζ )) ≤ , which is the desired inequality. Case II: a ◦ u ( ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ )) = { a (cid:48) , b (cid:48) } . We break this into two further sub-cases. Case IIa: (cid:82) ( a ◦ u ) − ( a (cid:48) ) ∩ ∂ (cid:101) S b (cid:48) a (cid:48) u ∗ λ ≥ r . In this case we see that (cid:101) S b (cid:48) a (cid:48) must be anannulus, with the image of one boundary component in { a (cid:48) } × M and the otherin { b (cid:48) } × M , furthermore by assumption (cid:82) ( a ◦ u ) − ( a (cid:48) ) ∩ ∂ (cid:101) S b (cid:48) a (cid:48) u ∗ λ ≥ r , and henceby the remarks immediately following the statements of properties (T1) - (T7),we may apply Proposition 4.30 (or Lemma 4.36 as appropriate), which preciselyguarantees that Area u ∗ g ( S r ( ζ )) ≤ , which is the desired inequality. Case IIb: (cid:82) ( a ◦ u ) − ( a (cid:48) ) ∩ ∂ (cid:101) S b (cid:48) a (cid:48) u ∗ λ ≤ r . This case has more in similarity withCase I than Case IIa, in the sense that we will estimate area directly rather thaninvoke Proposition 4.30. We begin by claiming that S r ( ζ ) ⊂ S b (cid:48) a (cid:48) ( ζ ); the proof isidentical to that of Case I. Moreover we have S r ( ζ ) ⊂ S b (cid:48) a (cid:48) ( ζ ) ⊂ S b (cid:48) a (cid:48) ∩ (cid:101) S b (cid:48) a (cid:48) ( ζ ) =: ˇ S, We also define (cid:98) S := S a (cid:48) −∞ ∩ (cid:101) S b (cid:48) a (cid:48) ( ζ )In this way we have S b (cid:48) −∞ ∩ (cid:101) S b (cid:48) a (cid:48) ( ζ ) = ˇ S ∪ (cid:98) S and ∂ ˇ S ∩ ∂ (cid:98) S = ∂ (cid:98) S, and in fact if we define Λ = ∂ (cid:101) S b (cid:48) a (cid:48) ( ζ ) ∩ ( a ◦ u ) − ( a (cid:48) )
22 J.W. FISH AND H. HOFER we then have (cid:101) S b (cid:48) a (cid:48) ( ζ ) ∩ ( a ◦ u ) − ( a (cid:48) ) = ∂ (cid:98) S ∪ Λ and ∂ (cid:98) S ∩ Λ = ∅ . With these preliminary definitions out of the way, we recall that by the hypothe-ses of Case IIb, we have (cid:90) Λ u ∗ λ ≤ r , and we aim to estimate that area of ˇ S , since S r ( ζ ) ⊂ ˇ S . Our technique will beto employ Theorem 9, but first we must estimate the quantity (cid:82) ∂ (cid:98) S ∪ Λ u ∗ λ . To thatend, we employ Lemma 4.37 and Theorem 9 in regards to (cid:82) ˜ α estimates to obtain, (cid:90) ∂ (cid:98) S u ∗ λ ≤ C h e C h − (cid:15) (cid:90) (cid:98) S u ∗ ω. Consequently, (cid:90) ( a ◦ u ) − ( a (cid:48) ) ∩ ∂S b (cid:48) a (cid:48) ( ζ ) u ∗ λ = (cid:90) ∂ (cid:98) S u ∗ λ + (cid:90) Λ u ∗ λ ≤ C h e C h − (cid:15) (cid:90) (cid:98) S u ∗ ω + 100 r ≤ C h e − (cid:90) (cid:98) S u ∗ ω + 100 r ≤ C h (cid:90) (cid:98) S u ∗ ω + 100 r (91)where the second inequality follows from equation (78), and the third inequalityfollows from the fact that e (2 − ) ≤ ≤
2. With this estimate in hand, we nowapply Theorem 9 to estimate the area of ˇ S .Area u ∗ g ( ˇ S ) = (cid:90) ˇ S u ∗ ( da ∧ λ + ω ) ≤ (cid:16) C − h (cid:90) Λ ∪ ∂ (cid:98) S u ∗ λ + (cid:90) ˇ S u ∗ ω (cid:17)(cid:0) e C h ( b (cid:48) − a (cid:48) ) − (cid:1) + (cid:90) ˇ S u ∗ ω ≤ (cid:16) C − h (cid:90) Λ ∪ ∂ (cid:98) S u ∗ λ + (cid:90) ˇ S u ∗ ω (cid:17)(cid:0) e C h − (cid:15) − (cid:1) + (cid:90) ˇ S u ∗ ω ≤ C − h (cid:90) Λ ∪ ∂ (cid:98) S u ∗ λ + 2 (cid:90) ˇ S u ∗ ω ≤ C − h (cid:16) C h (cid:90) (cid:98) S u ∗ ω + 100 r (cid:17) + 2 (cid:90) ˇ S u ∗ ω ≤ r C − h + 2 (cid:90) (cid:101) S b (cid:48) a (cid:48) u ∗ ω ≤ C − h + 12 ≤
120 + 12 ≤ , where the second inequality follows from equation (85), the third inequality followsfrom equation (78) and the fact that e (2 − ) ≤
2, the fifth inequality the fact thatˇ S ∪ (cid:98) S ⊂ (cid:101) S b (cid:48) a (cid:48) ⊂ S \ S a − a and property (L1), the sixth inequality follows fromthe fact that r ≤ as guaranteed by Lemma 4.29 and property (L1) again,and the seventh inequality follows from the definition of the ambient geometry Recall that (cid:101) α = − (˜ u ∗ da ) ◦ ˜ , and when the pseudoholomorphic map is unperturbed, wesimply have (cid:101) α = − ( u ∗ da ) ◦ j = u ∗ λ. ERAL CURVES AND MINIMAL SETS 123 constant provided in Definition 4.11. Recall that we have already established that S r ( ζ ) ⊂ ˇ S , and hence we haveArea u ∗ g ( S r ( ζ )) ≤ Area u ∗ g ( ˇ S ) ≤ , which is the desired estimate, and completes Case IIb.Since cases I, IIa, and IIb exhaust all possibilities, we conclude thatArea u ∗ g ( S r ( ζ )) ≤ ζ such that a ◦ u ( ζ ) ≥ a + 2. Recall we have defined the compact set K = [ − a − , a + 2] × M , and hence this completes the proof Theorem 5. (cid:3) Proof of Theorem 6: Asymptotic Curvature Bound.
The main purpose of this section is to prove the following result.
Theorem 6 (asymptotic curvature bound) . Let ( M, η ) be a compact framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . For each feral pseudoholomorphic curve u =( u, S, j, R × M, J, µ, D ) , there exists a compact set of the form K := [ − a , a ] × M ,and positive constant C κ = C κ ( M, η, J, g ) with the following significance. First, therestricted map u : S \ u − ( K ) → R × M is an immersion. Second, for each ζ ∈ S \ u − ( K ) we have (cid:107) B u ( ζ ) (cid:107) ≤ C κ where B u ( ζ ) is the second fundamental form of the immersion u evaluated at thepoint ζ .Proof. Suppose not. Then there exists a sequence of points ζ k ∈ S with the propertythat | a ◦ u ( ζ k ) | → ∞ for which either T u ζ k = 0 for all k ∈ N or else (cid:107) B u ( ζ k ) (cid:107) → ∞ .Without loss of generality, we will assume a ◦ u ( ζ k ) → ∞ monotonically; the case a ◦ u ( ζ k ) → −∞ is essentially identical. Recall that Theorem 5 guarantees thatthere exists an a > r = r ( M, η, J, g ) > u ∗ g (cid:0) S r ( ζ ) (cid:1) ≤ ζ ∈ S for which a ◦ u ( ζ ) ≥ a , and where S r ( ζ ) is the connected componentof u − ( B r ( u ( ζ ))) containing ζ ; here B r ( p ) denotes an open metric ball in R × M of radius r centered at p . By increasing a if necessary, we may also assume that a and − a are regular values of a ◦ u ,Genus( S ) = Genus (cid:0) u − (cid:0) [ − a , a ] × M (cid:1)(cid:1) , and Punct( S ) equals the number of non-compact path-connected components ofthe set S \ u − (cid:0) ( − a , a ) × M (cid:1) . With the abbreviation S a = u − (cid:0) ( − a , a ) × M (cid:1) we may also assume (cid:90) S \ S a u ∗ ω ≤ min (cid:16) , (cid:126) , − (cid:15) r , (2 (cid:15) − + 10 C h ) − (cid:17) (92)where (cid:126) = (cid:126) ( M, η, J, g, − (cid:15) ,
0) is the constant guaranteed by Theorem 4, and (cid:15) = min( C − h , r ) as in equation (78), C h is the ambient geometry constant given
24 J.W. FISH AND H. HOFER in Definition 4.11, and r is the positive constant provided in Lemma 4.29. Further-more, by passing to a subsequence if necessary we may assume a ◦ u ( ζ ) > a + 1, u ( µ ∪ D ) ∈ [ − a , a ] × M , and that a ◦ u ( ζ k +1 ) − a ◦ u ( ζ k ) ≥ r )(93)for all k ∈ N . For notational convenience, we define S k := S r ( ζ k ) , and we define the maps(94) v k : S k → [ − , × M given by v k ( ζ ) := Sh a ◦ u ( ζ k ) ◦ u ( ζ )where Sh ( · ) is the shift map Sh x : R × M → R × M Sh x ( a, p ) = ( a − x, p ) . Next observe that by construction v k ( ζ k ) ∈ { } × M for all k where M is com-pact. We conclude that after passing to a further subsequence, still denoted withsubscripts k , we have convergence of the sequence of points(95) v k ( ζ k ) → p := (0 , p (cid:48) ) ∈ { } × M, and B r ( p ) ⊂ B r ( v k ( ζ k )) , where B r ( p ) denotes the closed metric ball of radius r centered at p . For notationalconvenience we define W = ( − , × M . Next we observe that the sequence ofpseudoholomorphic curves ( v k , S k , j k , W, J, ∅ , ∅ ) have uniformly bounded area, zerogenus, ∂S k = ∅ , v − k ( B r ( p )) is compact, and v k ( ζ k ) → p . We conclude from The-orem 2.36 (target-local Gromov compactness) that after passing to a subsequence,still denoted with subscripts k , there exist compact surfaces with boundary (cid:101) S k ⊂ S k with the property that v k ( S k \ (cid:101) S k ) ⊂ W \ B r ( p ) and with the property that thepseudoholomorphic curves ˜ v k := (˜ v k , (cid:101) S k , ˜ j k , W, J, ∅ , ∅ )defined by ˜ v k = v k (cid:12)(cid:12) (cid:101) S k and ˜ j k = j k (cid:12)(cid:12) (cid:101) S k , converge in a Gromov sense to the pseu-doholomorphic curve ˜ v := (˜ v, (cid:101) S, ˜ j, W, J, ∅ , (cid:101) D ) . In particular, there will exist decorations ˜ r for the nodal points (cid:101) D ⊂ (cid:101) S and diffeo-morphisms φ k : (cid:101) S (cid:101) D, ˜ r → (cid:101) S k for which ˜ v k ◦ φ k → ˜ v in C ∞ loc ( (cid:101) S (cid:101) D, ˜ r \∪ i Γ i ) where the Γ i are the special circles obtainedby blowing up the nodal points, and ˜ v k ◦ φ k → ˜ v in C ( (cid:101) S (cid:101) D, ˜ r ). As a final consequenceof Theorem 2.36, we note that ˜ v is an immersion along ∂ (cid:101) S , and ˜ v ( ∂ (cid:101) S ) ∩B r ( p ) = ∅ .As a consequence of these facts, we see that p ∈ ˜ v ( (cid:101) S ). Moreover, we define ˆ ζ k ∈ (cid:101) S so that φ k (ˆ ζ k ) = ζ k ∈ S , and thus ˜ v k ◦ φ k (ˆ ζ k ) → p . If needed, we then pass to asubsequence so that ˆ ζ k → ˆ ζ ∞ ∈ (cid:101) S \ ∂ (cid:101) S . See Definition 2.35.
ERAL CURVES AND MINIMAL SETS 125
In what follows, it will be convenient to have a bit more control over the ∂ (cid:101) S k .As such, we choose a regular value r ∈ (0 , r ] of the function ρ : (cid:101) S → R ρ ( ζ ) = dist g (cid:0) p, ˜ v ( ζ ) (cid:1) , for which ˜ v ( (cid:101) D ) ∩ ∂ B r ( p ) = ∅ . We then define (cid:98) S ⊂ ˜ v − ( B r ( p )) to be the unionof connected components of ˜ v − ( B r ( p )) with the property that | (cid:98) S | := (cid:98) S/ ∼ isconnected and ζ ∞ ∈ (cid:98) S . This allows us to define the pseudoholomorphic curveˆ v = (cid:0) ˆ v, (cid:98) S, ˆ j, W, J, ∅ , (cid:98) D (cid:1) where ˆ v = ˜ v (cid:12)(cid:12) (cid:98) S , ˆ j = ˜ j (cid:12)(cid:12) (cid:98) S , and (cid:98) D = (cid:101) D ∩ (cid:98) S. We then define (cid:98) S k := φ k (cid:0) (cid:98) S (cid:98) D, ˜˜ r (cid:1) ⊂ (cid:101) S k so that for the pseudoholomorphic curvesˆ v k = (cid:0) ˆ v k , (cid:98) S k , ˆ j k , W, J, ∅ , ∅ (cid:1) defined from the ˜ v k via domain restriction, we have ˆ v k → ˆ v in a Gromov sense. Toproceed, we will need the following. Lemma 4.40 (properties of limit curve) . The limit curve (ˆ v, (cid:98) S, ˆ j, W, J, ∅ , (cid:98) D ) is not nodal; that is, (cid:98) D = ∅ . Moreover, the limitcurve is generally immersed in the sense of Definition 2.19. We will postpone the proof of Lemma 4.40 until a bit later because the proofdistracts from the main argument. For the moment then, we assume it is true,and hence (cid:98) S is connected. Observe that as a consequence of Gromov convergence,Lemma 4.40, and our construction of the ˆ v , it follows thatˆ v k ◦ φ k → ˆ v in C ∞ ( (cid:98) S, [ − , × M ) . Recall that as part of our argument to derive a contradiction, we have assumedthat the ζ k ∈ S have the property that | a ◦ u ( ζ k ) | → ∞ and either T u ( ζ k ) = 0 orelse (cid:107) B u ( ζ k ) (cid:107) → ∞ . We have also defined ˆ ζ k ∈ (cid:98) S so that φ k (ˆ ζ k ) = ζ k and ˆ ζ k → ˆ ζ ∞ .As a consequence of our above definitions, we then have:either T ˆ v k ( ζ k ) = 0 or else (cid:107) B ˆ v k ( ζ k ) (cid:107) → ∞ . Note that in either case, we must have T ˆ v (ˆ ζ ∞ ) = 0. Indeed, if T ˆ v (ˆ ζ ∞ ) (cid:54) = 0, then ˆ v is immersed in a neighborhood of ˆ ζ ∞ , and hence (cid:107) T ˆ v (cid:107) is bounded away from zeroin a neighborhood of ˆ ζ ∞ and (cid:107) B ˆ v (cid:107) is bounded in a neighborhood of ˆ ζ ∞ ; makinguse of the fact that ˆ ζ k → ˆ ζ ∞ and ˆ v k → ˆ v in C ∞ then would yield a contradiction.Thus we have shown that T ˆ v (ˆ ζ ∞ ) = 0 as claimed.Our next task is then to prove that in fact T ˆ v (ˆ ζ ∞ ) (cid:54) = 0, which would then yieldthe desired contradiction to prove Theorem 6. To that end, recall that we haveassumed that | a ◦ u ( ζ k ) | → ∞ , and by construction the (cid:98) S k ⊂ S are all pairwisedisjoint. Moreover, because (cid:82) S u ∗ ω < ∞ , and ω evaluates non-negatively on J -complex lines, it follows that (cid:82) (cid:98) S k ˆ v ∗ k ω →
0, and hence (cid:82) (cid:98) S ˆ v ∗ ω = 0. Also recall that Here ζ ∼ ζ (cid:48) for ζ (cid:54) = ζ (cid:48) if and only if { ζ, ζ (cid:48) } ⊂ (cid:101) D ∩ (cid:98) S forms a nodal pair.
26 J.W. FISH AND H. HOFER as a consequence of Lemma 4.40, ˆ v is generally immersed. From these facts weconclude the following about the image of ˆ v :ˆ v ( (cid:98) S ) ⊂ D := B r ( p ) ∩ (cid:0) [ − , × β (cid:0) [ − r , r ] (cid:1)(cid:1) where β is a solution to the differential equation β (cid:48) = X η ( β ) with (0 , β (0)) = p .We note that D is a holomorphically embedded disk. As a consequence we canfind a compact disk-like domain with smooth boundary D ⊂ C , which satisfies0 ∈ D \ ∂ D ⊂ C , supporting a holomorphic diffeomorphism of the form ψ : D → D given by ψ ( s, t ) = (cid:0) s, β ( t ) (cid:1) . We note that ψ is also an isometric embedding with respect to the flat metric ds + dt on C . Recall by construction thatˆ v (ˆ ζ ∞ ) = p ∈ D with T ˆ v (ˆ ζ ∞ ) = 0 , and hence ˆ v : (cid:98) S → D is a branched cover with ˆ ζ ∞ a branch point. A consequenceof target-local Gromov compactness, Theorem 2.36, is that the map ˆ v is immersedalong ∂ (cid:98) S , and by Lemma 2.4.1 of [30] it follows that the set of critical pointsof the map ˆ v is finite. Because of the latter, we will assume r > ζ ∞ is the unique critical point of ˆ v : (cid:98) S → D . Morespecifically, we follow the trimming procedure to obtain ˆ v from ˜ v but for which r chosen sufficiently small so as to meet our needs. In either case, we do notintroduce new notation to indicated this newly trimmed curve. As a consequenceof this construction, it is then an elementary exercise from complex variables toshow that there exists complex coordinates z on (cid:98) S so that(96) ψ − ◦ ˆ v ( z ) = z n † with n † ≥ . With this local patch of limit curve understood as a branched cover of a disk, weaim to use this structure and Gromov convergence, to back up in the sequence tostudy compact manifolds with boundary of the formΣ k := { ζ ∈ S : a ◦ u ( ζ k ) − c ≤ a ◦ u ( ζ ) ≤ a ◦ u ( ζ k ) + c } . for some small generic choice of c . Modulo the addition of some small “inessentialcapping disks” (defined below), and for n † ≥ n † -gon neighborhood Σ k, ⊂ Σ k of ˆ ζ k , which we will use toshow Σ k (or rather the capped surface (cid:101) Σ k ) has negative Euler characteristic forall sufficiently large k ∈ N , and hence S has either infinitely many ends (which isimpossible), or infinite genus (which is also impossible). This will yield the desiredcontradiction. It may be helpful to consider Figure 4.4.We now proceed with the details. Recalling the point p = (0 , p (cid:48) ) defined in (95),we begin by defining the set E ⊂ (cid:98) S by E := ˆ v − (cid:0) [0 , × { p (cid:48) } (cid:1) . We also fix c ∈ R so that 0 < c ≤ r with the property that the set { a ◦ u ( ζ k ) − c , a ◦ u ( ζ k ) + c } k ∈ N is contained in the set of regular values of the function a ◦ u : S → R . Note that since c (cid:54) = 0, we also have that ± c are regular values a ◦ ˆ v : (cid:98) S → R . We then define the compact surfaces with boundaryΣ k := { ζ ∈ S : a ◦ u ( ζ k ) − c ≤ a ◦ u ( ζ ) ≤ a ◦ u ( ζ k ) + c } . ERAL CURVES AND MINIMAL SETS 127
Figure 2. Σ k, ⊂ Σ k . Note that image of the two-dimensionalΣ k, lies in a four-dimensional space (which is difficult to draw).Note that the important sequence (ˆ ζ k ), which is not indicated inthis figure, consists of points close to the points φ k (ˆ ζ k ) . We also define important sub-surfaces of the Σ k in the following manner. Recallthat φ k : (cid:98) S → (cid:101) S k ⊂ S , so we may regard the φ k as having image in S . Since wealso have Σ k ⊂ S by construction, we then define the sequence of sets ˙ E k by˙ E k := φ k ( E ) ∩ ∂ Σ k . For all sufficiently large k ∈ N , we have by construction that the set ˙ E k consists of n † points, where n † is the natural number given in equation (96). We now definethe manifolds Ξ k := ∂ Σ k and equip them with the metric γ k = u ∗ g (cid:12)(cid:12) Ξ k . Define˙ F k ⊂ Σ k to be the set of points given by ˙ F k := { ξ ∈ Ξ k : dist γ k ( ξ, E k ) = r } .Observe that for all sufficiently large k ∈ N the sets ˙ F k consist of 2 n † points. Define L k ⊂ Σ k ⊂ S to be the (image of the) u ∗ g -gradient trajectories in Σ k terminatingin ˙ F k . Define Σ k, to be the closure of the connected component of Σ k \ L k whichcontains ζ k . Observe that by construction, for all k ∈ N sufficiently large Σ k, is a smooth manifold with piecewise smooth boundary and homeomorphic to aclosed disk. Furthermore, the single boundary component of Σ k, is comprised of4 n † smooth segments connected together at 4 n † corners. Moreover, 2 n † of thesesmooth segments are u ∗ g -gradient flow lines of the function a ◦ u , and n † segmentsare contained in the level set ( a ◦ u ) − ( a ◦ u ( ζ k )+ c ), and n † segments are containedin the level set ( a ◦ u ) − ( a ◦ u ( ζ k ) − c ). Later it will be useful to recall that foreach connected component Ξ (cid:48) of ( a ◦ u ) − ( a ◦ u ( ζ k ) ± c ) we have (cid:90) Ξ (cid:48) u ∗ λ ≥ r (97)for all sufficiently large k ∈ N .
28 J.W. FISH AND H. HOFER
Next we define a set ∆ k to consist of those compact connected components ∆ (cid:48) of the set S \ (Σ k \ ∂ Σ k ) which have empty intersection with ( a ◦ u ) − ([ − a , a ]).We call these inessential caps . We note that as a consequence of equation (92),equation (93), and Theorem 4, it follows that if ∆ (cid:48) ∈ ∆ k with Σ k ∩ ∆ (cid:48) (cid:54) = ∅ , then a ◦ u ( ζ k ) − c − ≤ inf ζ ∈ ∆ (cid:48) a ◦ u ( ζ ) < sup ζ ∈ ∆ (cid:48) a ◦ u ( ζ ) ≤ a ◦ u ( ζ k ) + c + 1 . (98)Consequently, each ∆ (cid:48) ∈ ∆ k has non-empty intersection with at most one of theΣ k (cid:48) , and thus we must have k (cid:48) = k . As in the proof of Theorem 5, we then define (cid:101) Σ k to be the union of Σ k with all those elements of ∆ k which have non-emptyintersection with Σ k . We now claim the following. Lemma 4.41 (inessential caps miss the 4 n -gon) . For all sufficiently large k ∈ N , we have Σ k, ∩ ( (cid:101) Σ k \ Σ k ) = ∅ . In other words, when k is large enough, the n † -gons Σ k, constructed above haveempty intersection with the inessential caps added to the Σ k to create (cid:101) Σ k . As above, we postpone the proof of Lemma 4.41 for now and complete the proofof Theorem 6. To that end, we now claim the following.
Lemma 4.42 (negative Euler characteristic) . Letting (cid:101) Σ k ( ζ k ) denote the connected component of (cid:101) Σ k containing ζ k , the followingholds: χ (cid:0)(cid:101) Σ k ( ζ k ) (cid:1) < where χ is the Euler characteristic. Again, we postpone the proof of Lemma 4.42 so as to complete the proof ofTheorem 6. Recall the following terminology from the proof of Theorem 5. We saythat two connected components L and L of ( a ◦ u ) − ( a ) are eventually connected provided that there exists a connected component ˇ S of S \ ( a ◦ u ) − (cid:0) ( − a , a ) (cid:1) forwhich L ∪ L ⊂ ˇ S . Thus we fix a > a sufficiently large so that for each pair ofconnected components L and L of ( a ◦ u ) − ( a ) which are eventually connected,there exists a connected component of ( a ◦ u ) − (cid:0) [ a , a ] (cid:1) which contains L ∪ L .With a established as in the proof of Theorem 5 (see after the proof of Lemma4.37) we now apply Lemma 4.38 and Lemma 4.39 which together guarantee thatfor all sufficiently large k we have that (cid:0) ∂ (cid:101) Σ k ( ζ k ) (cid:1) ≤ . That is, the number of connected components of ∂ (cid:101) Σ k ( ζ k ) is at most two. Howeverby super-additivity of genus , we also haveGenus (cid:0)(cid:101) Σ k ( ζ k ) (cid:1) = 0for all sufficiently large k . From these two observations, we deduce that χ (cid:0)(cid:101) Σ k ( ζ k ) (cid:1) ≥ , but this contradicts Lemma 4.42. This is the desired contradiction which completesthe proof of Theorem 6 (modulo the proofs of Lemma 4.40, Lemma 4.41, and Lemma4.42). (cid:3) See Lemma 2.29.
ERAL CURVES AND MINIMAL SETS 129
Proof of Lemma 4.42.
Recall that we must show that χ (cid:0)(cid:101) Σ k ( ζ k ) (cid:1) < (cid:101) Σ k ( ζ k )is the connected component of (cid:101) Σ k containing ζ k . For the sake of notational conve-nience, we define (cid:101) Σ (cid:48) k := (cid:101) Σ k ( ζ k ) and (cid:101) Σ (cid:48) k, := Σ k, , and denote by (cid:101) Σ (cid:48) k, , . . . , (cid:101) Σ (cid:48) k,n k the connected components of (cid:101) Σ (cid:48) k \ (cid:101) Σ (cid:48) k, . We will needthe following ad hoc definition. Definition 4.43 (surface with special boundary) . A surface with special boundary is a smooth compact real two-dimensional orientedmanifold S with piece-wise smooth boundary and zero genus, which additionally hasthe following properties. The boundary of S is the union of three sets denoted ∂ + S , ∂ − S , and ∂ S where(1) ∂ S is diffeomorphic to the disjoint union of finitely many compact inter-vals,(2) each of ∂ − S and ∂ + S is diffeomorphic to the disjoint union of finitely manycompact intervals and circles R / Z (3) ∂ − S ∩ ∂ + S = ∅ (4) each connected component of ∂ S intersects each of ∂ + S and ∂ − S exactlyonce(5) neither ∂ − S nor ∂ + S is empty. It is worth noting that each of (cid:101) Σ (cid:48) k, , (cid:101) Σ (cid:48) k, , . . . , (cid:101) Σ (cid:48) k,n k are surfaces with specialboundary. Next we need to understand the effect on the Euler characteristic ofgluing such surfaces along their “sides” ∂ S . This is accomplished via the following. Lemma 4.44 (cuts increase Euler characteristic) . Let S be a surface with special boundary as in Definition 4.43. Let L ⊂ S denote asmoothly embedded compact interval which transversely intersects each of ∂ − S and ∂ + S precisely once and for which L ∩ ∂ S = ∅ . Let Σ denote the surface with specialboundary obtained by cutting S along L . More precisely, this means we consider S \ L as a Riemannian manifold with boundary, equipped with a metric (that is,a distance function) essentially defined as the length of the shortest path in S \ L connecting a pair of points, and then we define Σ to be the metric closure of S \ L .In this way, we have S (cid:54) = Σ := S \ L = ( S \ L ) ∪ L ∪ L where each L i is diffeomorphic to L , and L ∪ L ⊂ ∂ Σ . Then χ (Σ) = χ ( S ) + 1 where χ ( M ) is the Euler characteristic of M .Proof. Recall the Gauss-Bonnet theorem for surfaces with boundary and corners,which states that χ ( S ) = 12 π (cid:16) (cid:90) S K g dA + (cid:90) ∂S κ g ds + n (cid:88) i =1 θ i (cid:17) where K g is the Gaussian curvature, κ g is the geodesic curvature, and the θ i areexternal angles at corners associated to a Riemannian metric g . To prove the
30 J.W. FISH AND H. HOFER
Figure 3.
Cutting S to obtain Σ.lemma, first choose a metric on S for which L , ∂ ± S , and ∂ S are all geodesics, andthen apply Gauss-Bonnet. (cid:3) We now observe that (cid:101) Σ (cid:48) k = (cid:101) Σ (cid:48) k, ∪ n k (cid:91) i =1 (cid:101) Σ (cid:48) k,i , where each of the (cid:101) Σ (cid:48) k, , . . . , (cid:101) Σ (cid:48) k,n k are special surfaces with boundary in the senseof Definition 4.43. Indeed, in this case we have ∂ ± (cid:101) Σ (cid:48) k,i = (cid:0) ∂ (cid:101) Σ (cid:48) k,i (cid:1) ∩ (cid:16) ( a ◦ u ) − (cid:0) a ◦ u ( ζ k ) ± c (cid:1)(cid:17) for i ∈ { , . . . , n k } , and ∂ (cid:101) Σ (cid:48) k,i consists of the remaining gradient-type boundarysegments. Moreover, we note that (cid:101) Σ (cid:48) k can be obtained by gluing the (cid:101) Σ (cid:48) k, , . . . , (cid:101) Σ (cid:48) k,n k components to (cid:101) Σ (cid:48) k, along appropriate gradient-type boundary segments. Observethat by construction, we have ∂ (cid:101) Σ (cid:48) k, ) = 2 n † where n † ≥
2, and n k ≤ n † . Alsonote that χ ( (cid:101) Σ (cid:48) k,i ) ≤ k ∈ N and i ∈ { , . . . , n k } . We then apply Lemma 4.44, which guaranteesthe following χ ( (cid:101) Σ (cid:48) k ) = n k (cid:88) i =0 χ ( (cid:101) Σ (cid:48) k,i ) − n † = 1 + n k (cid:88) i =1 χ ( (cid:101) Σ (cid:48) k,i ) − n † ≤ n k − n † ≤ − n † ≤ − . This completes the proof of Lemma 4.42. (cid:3)
Proof of Lemma 4.41.
Recall that we must show that for all sufficiently large k ∈ N ,we have Σ k, ∩ ( (cid:101) Σ k \ Σ k ) = ∅ . We note that as a consequence of our construction, ERAL CURVES AND MINIMAL SETS 131 specifically equation (98), we have (cid:16) sup ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) − (cid:16) inf ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) ≤ c )and consequently the (cid:101) Σ k are all pairwise disjoint. Since they are disjoint, we findthat as k → ∞ we have (cid:90) (cid:101) Σ k u ∗ ω → , and hence another application of Theorem 4, guarantees that (cid:16) sup ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) − (cid:16) inf ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) → c and(99) (cid:90) (cid:101) Σ k \ Σ k u ∗ ω → . Now we note by construction that if Σ k, ∩ ( (cid:101) Σ k \ Σ k ) (cid:54) = ∅ , then (cid:101) Σ k \ Σ k and Σ k, must overlap on a connected component of Σ k, ∩ ( a ◦ u ) − ( { a ◦ u ( ζ k ) ± c } ) ⊂ Σ k, .Because the integral of λ along such components tend to r , we can conclude that(100) (cid:90) ∂ ( (cid:101) Σ k \ Σ k ) u ∗ λ ≥ r for all sufficiently large k ; see for example equation (97). However, we then invokeTheorem 9, which guarantees that(101) (cid:90) ∂ ( (cid:101) Σ k \ Σ k ) u ∗ λ ≤ (cid:16) C h (cid:90) (cid:101) Σ k \ Σ k u ∗ ω + 0 (cid:17) e C h δ k where δ k := (cid:16) sup ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) − (cid:16) inf ζ ∈ (cid:101) Σ k a ◦ u ( ζ ) (cid:17) − c → . In light of equation (99), we see that equation (101) contradicts equation (100).This contradiction then guarantees that indeed,Σ k, ∩ ( (cid:101) Σ k \ Σ k ) = ∅ which completes the proof of Lemma 4.41. (cid:3) Proof of Lemma 4.40.
Recall that W = ( − , × M and that we must prove thatthe limit curve ˆ v = (ˆ v, (cid:98) S, ˆ j, W, J, ∅ , (cid:98) D )is not nodal; that is, that (cid:98) D = ∅ . To that end, we suppose not, and we will derive acontradiction. First however, we will need to briefly recall some facts about Gromovconvergence. In particular, the set of nodes is given by (cid:98) D = { d , d , . . . , d n d , d n d } ,with { d ν , d ν } ⊂ (cid:98) D a nodal pair. In particular, for each nodal pair { d ν , d ν } ⊂ (cid:98) D wehave ˆ v ( d ν ) = ˆ v ( d ν ). Next we recall that (cid:98) S (cid:98) D is defined to be the circle compactifi-cation of (cid:98) S \ (cid:98) D (or more specifically, an oriented blow-up at the points in (cid:98) D ), andthe newly added circles are denoted Γ ν and Γ ν , which signifies that each circle Γ ν is associated to a nodal point d ν and similarly for Γ ν and d ν . The surface (cid:98) S (cid:98) D, ˆ r is then obtained by gluing pairs of circles Γ ν and Γ ν via the orientation reversingorthogonal maps r ν : Γ ν → Γ ν ; here ˆ r = { r , . . . , r n d } is called a decoration. It is
32 J.W. FISH AND H. HOFER useful to let Γ ν ⊂ (cid:98) S (cid:98) D, ˆ r denote the circle obtained by by gluing Γ ν and Γ ν . Alsorecall that the definition of Gromov convergence guarantees the existence of diffeo-morphisms φ k : (cid:98) S (cid:98) D, ˆ r → (cid:98) S k with the property that φ ∗ k ˆ v k → ˆ v in C , φ ∗ k ˆ v k → ˆ v in C ∞ loc ( (cid:98) S r,D \ ∪ ν Γ ν ), φ ∗ k j k → j in C ∞ loc ( (cid:98) S (cid:98) D, ˆ r \ ∪ ν Γ ν ). In particular, this guaranteesthat there exists a sequence (cid:15) k → φ ∗ k ˆ v k (Γ ν ) ⊂ B (cid:15) k ( p ν ) , where p ν := ˆ v ( d ν ) = ˆ v ( d ν ) = ( a ν , q ν ) ∈ W. We note that by the construction of ˆ v , the p ν belong to W ; see the set-up beforethe initial statement of Lemma 4.40. Lemma 4.45 (some local properties) . Let ˆ v k = (ˆ v k , (cid:98) S k , ˆ j k , W, J, ∅ , ∅ ) and ˆ v = (ˆ v, (cid:98) S, ˆ j, W, J, ∅ , (cid:98) D ) be as above with ˆ v k → ˆ v in a Gromov sense, and let φ k : (cid:98) S (cid:98) D, ˆ r → (cid:98) S k be the associated diffeomorphisms, andlet { Γ , . . . , Γ n d } be the collection of circles Γ ν obtained by identifying Γ ν = Γ ν ;see above. Fix Γ ∈ { Γ , . . . , Γ n d } , and let (cid:98) Σ be the connected component of (cid:98) S (cid:98) D, ˆ r containing Γ . Then (cid:98) Σ \ Γ is disconnected with connected components given by (cid:98) Σ and (cid:98) Σ , and for all sufficiently large k ∈ N , there exists an (cid:15) > , ζ ∈ (cid:98) Σ , and ζ ∈ (cid:98) Σ such that a ◦ ˆ v k ◦ φ k ( ζ i ) − sup ζ ∈ Γ a ◦ ˆ v k ◦ φ k ( ζ ) ≥ (cid:15), (103) for each ζ i ∈ { , } .Proof. Recall, for example from the proof of Lemma 2.29, the removal of a loopfrom a surface either disconnects the surface or else reduces the genus. However,by construction Genus( (cid:98) S k ) = 0, so that the removal φ k (Γ) must disconnect (cid:98) Σ into (cid:98) Σ and (cid:98) Σ as required. To proceed, we make the following claim. Claim: ∂ (cid:98) S (cid:98) D, ˆ r ∩ (cid:98) Σ i (cid:54) = ∅ for each i ∈ { , } . To see this, we first let ˇ S be a connected component of (cid:98) S (cid:98) D, ˆ r \ ∪ ν Γ ν , and thenobserve that ˆ v : ˇ S → W is a pseudoholomorphic map, which is either a constantmap or generally immersed. Furthermore, because (cid:82) S u ∗ ω < ∞ and because the (cid:98) S k ⊂ S are disjoint, it follows that (cid:82) (cid:98) S k ˆ v ∗ k ω → (cid:82) ˇ S ˆ v ∗ ω = 0. As aconsequence of this, it follows that ˆ v ( ˇ S ) is contained in a patch of orbit cylinder.By unique continuation it then follows that ˆ v ( ˇ S ) is either a point in B r ( p ), orelse it has nontrivial intersection with ∂ B r ( p ); and recall that ˆ v − (cid:0) ∂ B r ( p (cid:1) ) = ∂ (cid:98) S .Thus if (cid:98) Σ i ∩ ∂ (cid:98) S (cid:98) D, ˆ r = ∅ for some i ∈ { , } then we must have that ˆ v restrictedto any connected component of ( (cid:98) S (cid:98) D, ˆ r \ ∪ k Γ k ) ∩ (cid:98) Σ i is a constant map. However,letting Σ i denote the image of (cid:98) Σ i under the quotient map (cid:98) S (cid:98) D, ˆ r → S/ ( d k ∼ d k ), wesee that ˆ v : Σ i → W must give rise to a compact stable pseudoholomorphic curve,with no marked points, zero (arithmetic) genus, on which ˆ v is constant on everycomponent; but this is impossible. We conclude that indeed, (cid:98) Σ i ∩ ∂ (cid:98) S (cid:98) D, ˆ r (cid:54) = ∅ . Thisestablishes the above claim. See Section 2.3 of [30].
ERAL CURVES AND MINIMAL SETS 133
To finish proving Lemma 4.45, we let { d, ¯ d } be the nodal pair associated to Γ,and we let ( a (cid:48) , q (cid:48) ) = ˆ v ( d ) = ˆ v ( ¯ d ) ∈ W . With ˆ v − (cid:0) ∂ B r ( p (cid:1) ) = ∂ (cid:98) S , { d, ¯ d } ∩ ∂ (cid:98) S = ∅ , (cid:82) (cid:98) Σ ˆ v ∗ ω = 0, and because (cid:98) Σ i ∩ ∂ (cid:98) S (cid:98) D, ˆ r (cid:54) = ∅ for each i ∈ { , } , it follows, since theimages of the maps ˆ v | (cid:98) Σ i are open in an orbit cylinder, that there exists an (cid:15) > a (cid:48) + 2 (cid:15), q (cid:48) ) ∈ (ˆ v ( (cid:98) Σ ) ∩ ˆ v ( (cid:98) Σ )) \ ∂ B r ( p ). Consequently, we define ζ ∈ (cid:98) Σ and ζ ∈ (cid:98) Σ by fixing ζ ∈ ˆ v − (cid:0) ( a (cid:48) + 2 (cid:15), q (cid:48) )) ∩ (cid:98) Σ and ζ ∈ ˆ v − (cid:0) ( a (cid:48) + 2 (cid:15), q (cid:48) )) ∩ (cid:98) Σ .Then by Gromov convergence, we haveˆ v k ◦ φ k ( ζ i ) → ( a (cid:48) + 2 (cid:15), q (cid:48) )for each i ∈ { , } . Also as a consequence of Gromov convergence, there exist (cid:15) k → v k ◦ φ k (Γ) ⊂ B (cid:15) k ( p (cid:48) )where p (cid:48) = ( a (cid:48) , q (cid:48) ) = ˆ v ( d ) = ˆ v ( ¯ d ). Inequality (103) then follows immediately. Thiscompletes the proof of Lemma 4.45. (cid:3) We are now prepared to complete the proof of Lemma 4.40. Indeed, as abovewe fix Γ ∈ { Γ , . . . , Γ n d } , and we will consider u − ([ a , ∞ ) × M ) \ ∪ ∞ k =1 φ k (Γ). Byconstruction, there exists sequences a k → ∞ and (cid:15) k → (cid:15) k > u ◦ φ k ( z ) ∈ [ a k − (cid:15) k , a k + (cid:15) k ] × M for all z ∈ φ k (Γ) . In view of (93) we have the inequality a k +1 − a k ≥
10 for large k . BecausePunct( S ) < ∞ and Genus( S ) < ∞ , and because the φ k (Γ) are all pairwise disjoint,it follows that only finitely many connected components of u − ([ a , ∞ ) × M ) \∪ ∞ k =1 φ k (Γ) have closure which is non-compact, and infinitely many which havecompact closure. We denote this infinite set of compact closures by { ˇΣ k (cid:48) } k (cid:48) ∈ N ,and observe that by construction it is the case that for each k (cid:48) ∈ N we have ∂ ˇΣ k (cid:48) ⊂ ∪ ∞ k =1 φ k (Γ). In fact, for each k (cid:48) ∈ N there exists a finite set F k (cid:48) ⊂ N suchthat ∂ ˇΣ k (cid:48) = ∪ k ∈ F k (cid:48) φ k (Γ). By virtue of the ˇΣ k (cid:48) being compact and with boundarycontained in ∪ ∞ k =1 φ k (Γ), the application of Lemma 4.45 guarantees not only thatthe function a ◦ u has an interior absolute maximum on each ˇΣ k (cid:48) , but also that forall sufficiently large k (cid:48) the maximal value of a ◦ u over ˇΣ k (cid:48) is at least some uniformthreshold amount larger than the maximal value of a ◦ u along the boundary. Moreprecisely, there exists an (cid:15) > k (cid:48) such that (cid:16) sup ζ ∈ ˇΣ k (cid:48) a ◦ u ( ζ ) (cid:17) − (cid:16) sup ζ ∈ ∂ ˇΣ k (cid:48) a ◦ u ( ζ ) (cid:17) ≥ (cid:15). Indeed, the uniformity of this inequality follows from Lemma 4.45. Now it followsfrom Theorem 4 that there exists an (cid:126) > (cid:82) ˇΣ k (cid:48) u ∗ ω ≥ (cid:126) for all sufficientlylarge k (cid:48) ∈ N , which then implies that (cid:82) S u ∗ ω = ∞ , which is impossible. This is thedesired contradiction which proves Lemma 4.40. (cid:3) Proof of Theorem 7: Existence Workhorse.
This section is devoted to the proof of Theorem 7. The main argument is pro-vided in Section 4.5.2, however this relies on some preliminary notions establishedin Section 4.5.1, and two technical results which are then proved in Section 4.5.3and Section 4.5.4.
34 J.W. FISH AND H. HOFER
Preliminaries.
Recall that a gradient flow line of a smooth, but possiblydegenerate, real-valued function defined on a closed manifold N need not have aunique point as its ω -limit set. That is to say, in general it may be the case thatfor a gradient flow line γ : R → N there exist sequences of real numbers t k → ∞ and t (cid:48) k → ∞ for which γ ( t k ) → p , γ ( t (cid:48) k ) → p (cid:48) , and p (cid:54) = p (cid:48) . Nevertheless, both p and p (cid:48) will be critical points of the associated function. This phenomenon isalso known for finite energy pseudoholomorphic curves, see [33]. Analogously, feralcurves need not have unique limits, but nevertheless by passing to a subsequenceone can extract the desired limit set which is indeed closed and invariant under theflow of X η . Here we make this precise with Definition 4.46 and Proposition 4.47below. Definition 4.46 ( x -limit set) . Let ( M, η ) be a closed framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . Let u = ( u, S, j, W, J, µ, D ) be a feral curvein the sense of Definition 1.5. For each x ∈ R , define (cid:98) Ξ x = Sh x (cid:16) (( x − , x + 1) × M ) ∩ u ( S ) (cid:17) ⊂ ( − , × M, where for each x ∈ R , the map Sh x : R × M → R × M is the shift map defined by Sh x ( a, p ) = ( a − x, p ) . Let x = { x i } i ∈ N ⊂ R be a monotonic sequence with either lim i →∞ x i = ∞ or lim i →∞ x i = −∞ . We then define the x -limit set of u to be thefollowing: L x := ∞ (cid:92) k =1 cl (cid:16) ∞ (cid:91) i = k (cid:98) Ξ x i (cid:17) ⊂ ( − , × M Proposition 4.47 (properties of x -limit set) . Let ( M, η ) be a closed framed Hamiltonian manifold, and let ( J, g ) be an η -adaptedalmost Hermitian structure on R × M . Let u = ( u, S, j, W, J, µ, D ) be a feral curveand x = { x i } i ∈ N ⊂ R be a monotonic sequence with | x i | → ∞ . Then the x -limit of u has the form ( − , × Ξ , where Ξ ⊂ M is a closed set which is invariant underthe Hamiltonian flow of η .Proof. The main technical tool to prove this result will be the following.
Lemma 4.48 (local invariance) . Let ( M, η ) , ( J, g ) , u , and x = { x i } i ∈ N be as above in Proposition 4.47. Then thereexists an (cid:15) > with the following property. If ( a , p ) ∈ L x , and if | (cid:15) | < (cid:15) , then ( a + (cid:15), p ) ∈ L x whenever | a + (cid:15) | < . Similarly if ( a , p ) ∈ L x , and if | (cid:15) | < (cid:15) ,then ( a , ϕ (cid:15)η ( p )) ∈ L x , where ϕ (cid:15)η is the time (cid:15) flow of the Hamiltonian vector fieldassociated to η . We will prove Lemma 4.48 momentarily, however for the moment we use it tocomplete the proof of Proposition 4.47. To that end, observe that Lemma 4.48 im-mediately establishes that L x = ( − , × Ξ with Ξ invariant under the Hamiltonianflow associated to η . Furthermore, by definition, L x is the intersection of closedsets, and hence itself closed. It is then elementary to deduce that Ξ is closed in M .This completes the proof of Proposition 4.47. (cid:3) Proof of Lemma 4.48.
We prove the case that x i → ∞ ; the case that x i → −∞ isessentially the same. To begin, we define (cid:15) := r where r = r ( M, η, J, g ) > ERAL CURVES AND MINIMAL SETS 135 bound). We then suppose that ( a , p ) ∈ L x , and q = ϕ (cid:15)η ( p ) for some | (cid:15) | < (cid:15) .By definition of L x , there exists a sequence { ζ k } k ∈ N ⊂ S and monotonic sequence { i k } k ∈ N ⊂ N with i k → ∞ for which Sh x ik ◦ u ( ζ k ) → ( a , p ). Note that since i k → ∞ , we must have x i k → ∞ .We then define a sequence of pseudoholomorphic curves u k = ( u k , (cid:101) S k , j k , R × M, J, ∅ , ∅ )where (cid:101) S k := S r ( ζ k ) ⊂ u − (cid:0) B r ( u ( ζ k )) (cid:1) ⊂ S is the connected component of u − (cid:0) B r ( u ( ζ k )) (cid:1) containing ζ k , and where j k := j (cid:12)(cid:12) (cid:101) S k and u k := Sh x ik ◦ u ; here B r ( p ) ⊂ R × M denotes the open metric ball of radius r centered at p . Note that by construction we have ζ k ∈ (cid:101) S k for every k ∈ N , and u k ( ζ k ) → ( a , p ). By construction, we may also apply Theorem 5 (asymptoticconnected-local area bound), which guarantees thatArea u ∗ k g ( (cid:101) S k ) ≤ . Also recall that u is a feral curve, and hence has finite genus, so by genus super-additivity and the fact that (cid:101) S k ⊂ S , it follows that Genus( (cid:101) S k ) is uniformlybounded in k . Also because u is feral it follows that µ ∪ D ) < ∞ , and x i k → ∞ sothat for all sufficiently large k we have ( µ ∪ D ) ∩ (cid:101) S k = ∅ , and hence the u k are stablefor all sufficiently large k . With uniform area bounds, uniform genus bounds, andstability, it then follows from Theorem 2.36, target-local Gromov compactness, thatafter passing to a subsequence (still denoted with subscripts k ), there exist com-pact Riemann surfaces with smooth boundary (cid:98) S k ⊂ (cid:101) S k which satisfy the followingproperties.(1) ζ k ∈ (cid:98) S k for all i ,(2) u k ( ζ k ) → ( a , p ),(3) u k ( ∂ (cid:98) S k ) ∩ B r / (( a , p )) = ∅ (4) ( u k , (cid:98) S k , j k , R × M, J, ∅ , ∅ ) → ( u ∞ , (cid:98) S ∞ , j ∞ , R × M, J, ∅ , D ∞ ) in a Gromovsense, where the limit is a compact pseudoholomorphic curve with immersedboundary.Furthermore, we note that because u : S → R × M is a feral curve, it follows that (cid:82) S u ∗ ω < ∞ , and because ω evaluates non-negatively on J -invariant planes andbecause a ◦ u ( ζ k ) ≥ x i k − → ∞ , it follows that (cid:82) S k u ∗ k ω →
0, and hence u ∗∞ ω = 0.Recall that ker ω = Span( ∂ a , X η ). We conclude that u ∞ ( (cid:98) S ∞ ) is contained in R × Γwhere Γ is the finite union of trajectories of the Hamiltonian vector field X η .Letting Γ p denote the Hamiltonian trajectory containing p , we note from thefact that u k ( ζ k ) → ( a , p ) and by definition of Gromov convergence that thereexists a connected component (cid:98) S (cid:48)∞ ⊂ (cid:98) S ∞ for which u ∞ ( (cid:98) S (cid:48)∞ ) ⊂ R × Γ p . Moreover,( a , p ) ∈ u ∞ ( (cid:98) S (cid:48)∞ ) ⊂ R × Γ p and u ∞ ( ∂ (cid:98) S (cid:48)∞ ) ∩ B r / (( a , p )) = ∅ , from which itfollows that for each q = ϕ (cid:15)η ( p ) with | (cid:15) | < r = (cid:15) we have ( a , q ) ∈ u ∞ ( (cid:98) S (cid:48)∞ ).But then by Gromov convergence, it follows that there exists a sequence ζ (cid:48) k ∈ (cid:98) S k such that u k ( ζ (cid:48) k ) → ( a , q ), and hence the sequence ζ (cid:48) k ∈ S satisfies x i k − ≤ a ◦ u ( ζ (cid:48) k ) ≤ x i k + 1 for all sufficiently large k ∈ N , so that ( a , q ) ∈ L x as required. See Lemma 2.29.
36 J.W. FISH AND H. HOFER
A similar argument shows that ( a + (cid:15), p ) ∈ L x whenever | (cid:15) | < (cid:15) , and | a + (cid:15) | < (cid:3) Main argument.
In what follows, it may be useful to review the notion of amarked nodal pseudoholomorphic curve as provided in Definition 2.30, as well asthe notion of a marked nodal Riemann surface as provided in Definition 2.20. Thelatter specifically is expressed as (
S, j, µ, D ) where D = { d , d , d , d , . . . } is theset of nodal points. Furthermore, as discussed after Remark 2.21, a (marked) nodalRiemann surface gives rise to the topological space | S | obtained by identifying eachpoint in D with its corresponding nodal pair; in other words | S | = S/ ( d i ∼ d i ). Weare now prepared to re-state the result we aim to prove here. Theorem 7 (existence workhorse) . Let ( M, η ) be a compact framed Hamiltonian manifold with dim( M ) = 3 . Let { a k } k ∈ N ⊂ R − be a sequence for which a k → −∞ monotonically. For each k ∈ N ,let ( J k , g k ) be a η -adapted almost complex structure on R × M . Suppose that thereexists a positive constant C ≥ , and suppose that for each k ∈ N and each b ∈ [ a k , there exists a stable unmarked but possibly nodal pseudoholomorphic curve u bk = (cid:0) u bk , S bk , j bk , ( −∞ , × M, J k , ∅ , D bk (cid:1) with the following properties.(P1) the topological space | S bk | is connected (implying (P4) below),(P2) u bk is compact and u bk ( ∂S bk ) ⊂ (0 , × M ,(P3) inf ζ ∈ S bk a ◦ u bk ( ζ ) = b ,(P4) there exists a continuous path α : [0 , → | S bk | satisfying a ◦ u bk ◦ α (0) = b and α (1) ∈ ∂S bk , (P5) Genus( S bk ) ≤ C ,(P6) (cid:82) S bk ( u bk ) ∗ ω ≤ C ,(P7) D bk ≤ C ,(P8) the number of connected components of ∂S bk is bounded above by C .Furthermore, suppose that J k → ¯ J in C ∞ , and for each fixed k , and each pair b, b (cid:48) ∈ [ a k , with b (cid:54) = b (cid:48) we have (cid:0) u bk ( S bk ) ∩ u b (cid:48) k ( S b (cid:48) k ) (cid:1) ≤ C. Then there exists a closed set Ξ ⊂ M satisfying ∅ (cid:54) = Ξ (cid:54) = M which is invariantunder the flow of the Hamiltonian vector field X η . Before we prove this theorem, we illustrate the hypotheses with an example. Weconsider for (
M, η ) the manifold R × M equipped with ( J k , g k ). Assume that forfixed k there exists a family of embedded J k -holomorphic disks with boundariesin (0 , × M . We assume that any two different disks in the family do not intersect,the image of a disk in the family lies in ( −∞ , × M and the set of minimum a -values covers [ a k , ω -energy bound. Thus we assume that we have such a sequence of Here we mean stable in the sense described in Definition 2.31. This is a geometric count of the intersection points of the images. It does not involvemultiplicities. It is not assumed to be a continuous family!
ERAL CURVES AND MINIMAL SETS 137
Figure 4.
For every k the figure shows a schematic family of J k -holomorphic embedded mutually disjoint disks whose minimal R -projection covers the interval [ a k , k , with the additional property that a k → −∞ as k → ∞ .That is, as we progress through the sequence, the associated families extend moreand more deeply into the negative end of R × M . Note that one can produce sucha system of disks in the case in which we have an exact symplectic cobordismfrom an overtwisted contact manifold M + on top to ( M, η ) on bottom. One canthen use Bishop’s theorem on disk fillings to construct the families. A completediscussion of Bishop’s theorem can be found in [1]. Of course, these ideas must becombined with the constructions and estimates derived in the current manuscript.Allowing non-embedded curves and mutual intersections increases the complexityof the argument. However, the basic idea can be seen in the special example whichwe have just outlined.
Proof.
We proceed via a proof by contradiction, and thus we begin by assumingTheorem 7 is false. Our first step in deriving a contradiction is then to fix c ≥ W k := (cid:0) k − , − a k (cid:1) × M,S k,c := S a k + ck (104) (cid:98) S k,c := (Sh a k ◦ u a k + ck ) − ( W k ) ⊂ S k,c , (105)where for each x ∈ R the shift map Sh x is defined bySh x : R × M → R × M Sh x ( a , q ) = ( a − x, q ) . We then define pseudoholomorphic curves(106) w k,c = (cid:0) w k,c , (cid:98) S k,c , j k,c , W k , J k , ∅ , (cid:98) D k,c (cid:1) where (cid:98) D k,c = D a k + ck ∩ (cid:98) S k,c One should be able to remove the exactness assumption, see Remark 1.2.
38 J.W. FISH AND H. HOFER j k,c = j a k + ck (cid:12)(cid:12) (cid:98) S k,c w k,c = Sh a k ◦ u a k + ck . (107)Observe that by definition, we also see that there exists a continuous path of theform α : [0 , → | (cid:98) S k,c | satisfying a ◦ w k,c ◦ α (0) = c and a ◦ w k,c ◦ α (1) ≥ − a k = | a k | → ∞ . (108)Moreover, each w k,c : (cid:98) S k,c → W k = ( k − , − a k ) × M ⊂ ( − , − a k ) × M is a propermap with empty intersection with ( −∞ , × M . To put this more geometrically,observe that (cid:98) S k,c ⊂ S k,c \ ∂S k,c and therefore w k,c : (cid:98) S k,c → W k has a naturalextension to w k,c : S k,c → ( − , ∞ ) × M via w k,c = Sh a k ◦ u a k ,ck . Consequently wemay regard the set-wise boundary ∂ (cid:98) S k,c ⊂ S k,c \ ∂S k,c , in which case we havesup ζ ∈ ∂ (cid:98) S k,c a ◦ w k,c ( ζ ) = inf ζ ∈ ∂ (cid:98) S k,c a ◦ w k,c ( ζ ) = − a k = | a k | → ∞ . (109)For later use, we also define the nodal Riemann surface ( S k,c , j k,c , D k,c ) by letting S k,c = S a k + ck as above and letting D k,c = D a k + ck ∩ S k,c . Recall that because the( J k , g k ) are η -adapted almost Hermitian structures, it follows from Lemma 2.7 thatthe ( W, J k , g k ) are indeed almost Hermitian manifolds with the g k expressible as g k := da ⊗ da + λ ⊗ λ + ω ( · , J k · ) . Also recall that by construction, the triples ( W k , J k , g k ) properly exhaust the al-most Hermitian manifold ( − , ∞ ) × M ⊂ R × M in the sense of Definition 2.37.Furthermore for each k ∈ N , the curve w k,c is a proper pseudoholomorphic curve in( W k , J k ), which can be included into R × M . Moreover, the symplectization coordi-nate of each w k,c has absolute minimum of c ; in other words, inf ζ ∈ S k,c a ◦ w k,c ( ζ ) = c .We now apply Theorem 3, which guarantees the existence of a sequence of positiveconstants C n , with the property that for every k ≥ n ∈ N we haveArea g k (cid:0) (cid:98) S nk,c (cid:1) = (cid:90) (cid:98) S nk,c w ∗ k,c ( da ∧ λ + ω ) ≤ C n where (cid:98) S nk,c := w − k,c ( W n ) ⊂ (cid:98) S k,c . We then apply Theorem 2.39 (exhaustive Gromov compactness), which guaranteesthe existence of a proper stable nodal pseudoholomorphic curve without boundary( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , to which a subsequence of the w k,c converge. We now make the following claim. Proposition 4.49 (feral limit curves) . Let c ≥ and let ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , be an exhaustive limit of some subsequence of the w k,c . Then ¯ w c is a feral pseudo-holomorphic curve in the sense of Definition 1.5. We postpone the proof of Proposition 4.49 until Section 4.5.3 below, and assumeits validity for the moment in order to complete the proof of Theorem 7. To thatend, we still aim to derive a contradiction, and thus we will need the followingresult.
ERAL CURVES AND MINIMAL SETS 139
Lemma 4.50 (bounded transverse intersections) . Consider non-negative numbers c, c (cid:48) ≥ with c (cid:48) > c , and let k (cid:55)→ (cid:96) k ∈ N be a strictlyincreasing sequence for which w (cid:96) k ,c → ¯ w c and w (cid:96) k ,c (cid:48) → ¯ w c (cid:48) in an exhaustive sense.Then the subset P ⊂ R × M of transversal intersection points of the two curves,which is defined by P := (cid:8) p ∈ R × M : there exists ( ζ, ζ (cid:48) ) ∈ S c × S c (cid:48) such that ¯ w c ( ζ ) = p = ¯ w c (cid:48) ( ζ (cid:48) ) and T ¯ w c ( ζ ) (cid:116) T ¯ w c (cid:48) ( ζ (cid:48) ) (cid:9) , satisfies P ≤ C. As before, we will postpone this proof until Section 4.5.4 below, and in themeantime proceed with the proof of Theorem 7. We now construct a sequence offeral curves in R × M . We start with the sequence of pseudoholomorphic curvesgiven by { w (cid:96), } (cid:96) ∈ N , and pass to a subsequence so that this subsequences converge(in an exhaustive Gromov sense) to the feral limit curve ¯ w . We will need to keeptrack of the subsequence in N which yields convergence, and thus we write w k ν , → ¯ w as ν → ∞ . We then consider the sequence of curves given by { w k ν , } ν ∈ N . We pass to a furthersubsequence to the obtain exhaustive Gromov convergence w k ν , → ¯ w . We then consider the sequence of curves given by { w k ν , } ν ∈ N , and we pass to afurther subsequence to obtain exhaustive Gromov convergence w k ν , → ¯ w . In this way we pass to further and further subsequences and obtain a sequence ofconverging sequences: w k (cid:96)ν ,(cid:96) → ¯ w (cid:96) for each (cid:96) ∈ N . We then pass to the diagonal subsequence, ¯ k ν := k νν , which by definition has theproperty that w ¯ k ν ,(cid:96) → ¯ w (cid:96) for each (cid:96) ∈ N . Recalling our notation, we then have¯ w (cid:96) = ( ¯ w (cid:96) , S (cid:96) , ¯ j (cid:96) , R × M, J, ∅ , D (cid:96) ) . We introduce another sequence of pseudoholomorphic curves denoted v (cid:96) = (cid:0) v (cid:96) , Σ (cid:96) , (cid:96) , ( − , × M, J, ∅ , ∆ (cid:96) (cid:1) and defined by Σ (cid:96) := ¯ w − (cid:96) (cid:0) ( (cid:96) − , (cid:96) + 1) × M (cid:1) v (cid:96) := Sh (cid:96) ◦ ¯ w (cid:96) (cid:96) := j (cid:96) (cid:12)(cid:12) Σ (cid:96) ∆ (cid:96) := D (cid:96) ∩ Σ (cid:96) . Note that this a bound on the number of intersection points of the images and not thenumber of parametrizing pairs.
40 J.W. FISH AND H. HOFER
By construction, the v (cid:96) are proper curves in ( − , × M , and they have uniformlybounded area and genus. Consequently, by Theorem 2.36, target-local Gromovcompactness, we can pass to a subsequence v (cid:96) ν and find compact domains (cid:101) Σ (cid:96) ⊂ Σ (cid:96) so that we have Gromov convergence (cid:0) v (cid:96) ν , (cid:101) Σ (cid:96) ν , (cid:96) ν , ( − , × M, J, ∅ , (cid:101) ∆ (cid:96) ν (cid:1) → ( v, Σ , , ( − , × M, J, ∅ , ∆)as ν → ∞ ; here (cid:101) ∆ (cid:96) ν = (cid:101) Σ (cid:96) ν ∩ ∆ (cid:96) ν . Also recall that these (cid:101) Σ (cid:96) ν have the property that v − (cid:96) ν (cid:0) [ − , ] × M (cid:1) ⊂ (cid:101) Σ (cid:96) ν . We also define the sets, { (cid:98) Ξ (cid:96) ν } ν ∈ N , by (cid:98) Ξ (cid:96) ν := Sh (cid:96) ν (cid:16)(cid:0) ( (cid:96) ν − , (cid:96) ν + 1) × M (cid:1) ∩ ¯ w ( S ) (cid:17) . We now make the following observation. Let (cid:96) (cid:48) ν be a subsequence of (cid:96) ν ; then forany such subsequence, the set (cid:98) Ξ := ∞ (cid:92) k =1 cl (cid:16) ∞ (cid:91) ν = k (cid:98) Ξ (cid:96) (cid:48) ν (cid:17) is the x -limit set L x of ¯ w : S → R × M for x = { a (cid:96) (cid:48) ν } ν ∈ N , in the sense of Definition4.46. By Proposition 4.47, we have L x = (cid:98) Ξ = ( − , × Ξ, where Ξ ⊂ M is a closedset which is invariant under the flow of the Hamiltonian vector field associated to η . At this point there are then three possible cases, where (cid:98) Ξ = L x : Case I: (cid:98)
Ξ = ∅ .Note, however, that this case is impossible because the curve ¯ w is proper with-out boundary but not compact and has image contained in [0 , ∞ ) × M . Indeed,properness follows as a consequence of ¯ w being feral, and it has image containedin [0 , ∞ ) × M as a consequence of exhaustive compactness together with the factthat the approximating curves w ¯ k ν , are a subsequence { w (cid:96), } (cid:96) ∈ N , and each w (cid:96), has image contained in [0 , ∞ ) × M by definition. To see that the curves are withoutboundary and not compact, it is sufficient to recall properties of the approximat-ing curves, { w ¯ k ν , } ν ∈ N ⊂ { w (cid:96), } (cid:96) ∈ N , and specifically the properties expressed inequation (108) and equation (109) together with the definition of exhaustive com-pactness. Case II: ∅ (cid:54) = (cid:98) Ξ (cid:54) = M .This case ruled out by our contradiction hypothesis. Case III: (cid:98)
Ξ = M .We assume this to be true for the remainder of our proof and seek to derive acontradiction. In fact, our contradiction hypothesis allows us to assume muchmore, namely that for each subsequence { (cid:96) (cid:48) ν } ν ∈ N of { (cid:96) ν } ν ∈ N we must have (cid:98) Ξ = M for the corresponding x -limit set (cid:98) Ξ. To proceed, let us define v (cid:96) ν = (cid:0) v (cid:96) ν , (cid:101) Σ (cid:96) ν , (cid:96) ν , ( − , × M, J, ∅ , (cid:101) ∆ (cid:96) ν (cid:1) and v = ( v, Σ , , ( − , × M, J, ∅ , ∆) , ERAL CURVES AND MINIMAL SETS 141 and recall from above that v (cid:96) ν → v in a Gromov sense as ν → ∞ . We then choosea finite set of points Z ⊂ Σ \ (∆ ∪ ∂ Σ), with the property that each point in Z isan immersed point of v , v ( z ) (cid:54) = v ( z (cid:48) ) for each z, z (cid:48) ∈ Z with z (cid:54) = z (cid:48) , and for each z ∈ Z we have v ∗ ω ( z ) (cid:54) = 0, and Z > C . Such a set Z exists as a consequenceof target-local Gromov compactness and properties of the approximating curves;specifically, a ◦ v has an absolute minimum of 0, and inf ζ ∈ ∂ (cid:101) Σ a ◦ v ( ζ ) ≥ . Wethen let Q ⊂ Σ denote the union of pairwise disjoint disk-like neighborhoods ofthe points in Z , each of which contains precisely one element of Z . We assumethat these disk-like neighborhoods are chosen so small that v : Q → R × M is anembedding. By Gromov convergence, there exist exist maps φ (cid:96) ν : Q → (cid:101) Σ (cid:96) ν forwhich(110) v (cid:96) ν ◦ φ (cid:96) ν → v in C ∞ ( Q, R × M ) . Next we note that by assumption (to derive a contradiction) it follows that afterpassing to a subsequence of the (cid:96) ν , denoted (cid:96) (cid:48) ν , there exists a sequence of finite sets { Z ν } ν ∈ N with Z ν ⊂ S for each ν ∈ N with the property that Sh (cid:96) (cid:48) ν ◦ ¯ w ( Z ν ) → v ( Z );it may be helpful to recall that S is the domain of the curve ¯ w . We then let r > { P ν } ν ∈ N with P ν ⊂ ¯ w − (cid:16) (cid:91) z ∈ Z ν B r ( ¯ w ( z )) (cid:17) ⊂ S , and with the property that each connected component of each P ν has non-emptyintersection with Z ν . Note that by shrinking r if necessary, we may assume thatthe B r (cid:0) ¯ w ( z ) (cid:1) are pairwise disjoint, and hence each connected component of P ν contains exactly one element of Z ν , and that for all sufficiently large ν ∈ N thenumber of connected components of P ν equals Z . We note that from Theorem5 it follows that Area ¯ w ∗ g ( P ν ) is uniformly bounded independent of ν , and the P ν have uniformly bounded genus since they are all subsets of S . We conclude fromTheorem 2.36, namely target-local Gromov compactness, that after passing to asubsequence, denoted with subscripts (cid:96) (cid:48) ν k , there exist compact Riemann surfaceswith boundary (cid:101) P ν k ⊂ P ν k with the property thatSh (cid:96) (cid:48) νk ◦ ¯ w ( ∂ (cid:101) P ν k ) ∩ (cid:91) z ∈ Z B r (cid:0) v ( z ) (cid:1) = ∅ , while Z ν k ⊂ (cid:101) P ν k \ ∂ (cid:101) P ν k and Sh (cid:96) (cid:48) νk ◦ ¯ w ( Z ν k ) → v ( Z ) , and furthermore we can arrange to have Gromov convergence of the maps(Sh (cid:96) (cid:48) νk ◦ ¯ w , (cid:101) P ν k , ¯ j , R × M, J, ¯ µ ∩ (cid:101) P ν k , D ∩ (cid:101) P ν k ) → ( ˇ w, (cid:101) P , ˇ j, R × M, J, ˇ µ, ˇ D ) . Importantly, by Theorem 6 (asymptotic curvature bound), the mapsSh (cid:96) (cid:48) νk ◦ w : (cid:101) P (cid:96) (cid:48) νk → R × M are immersions with uniformly bounded curvature. We conclude that that ˇ D = ∅ ,and ˇ w : (cid:101) P → R × M is an immersion, and, by Gromov convergence, there existembeddings ϕ k : (cid:101) P → (cid:101) P ν k with the property that(111) Sh (cid:96) (cid:48) νk ◦ ¯ w ◦ ϕ k → ˇ w in C ∞ ( (cid:101) P , R × M ) .
42 J.W. FISH AND H. HOFER
Note that by shrinking r if necessary, we may assume in fact that ˇ w is an embed-ding. Moreover, by construction v ( Z ) ⊂ ˇ w ( (cid:101) P ), and, because ∪ k ∈ N (cid:101) P ν k ⊂ S and (cid:82) S ¯ w ∗ ω < ∞ , it follows that ˇ w ∗ ω ≡
0. We conclude that ˇ w : (cid:101) P → R × M and v : Q → R × M transversally intersect at v ( Z ); indeed, this follows from the factthat ˇ w ∗ ω ≡ v ∗ ω (cid:54) = 0. Then by equation (110) and equation (111) andLemma 4.51, it follows that for all sufficiently large k the mapsSh (cid:96) (cid:48) νk ◦ ¯ w ◦ ϕ k : (cid:101) P → R × M and v (cid:96) (cid:48) νk ◦ φ (cid:96) (cid:48) νk : Q → R × M intersect transversally at Z > C points. However, by definition of the v (cid:96) (cid:48) νk itfollows that the maps Sh (cid:96) (cid:48) νk ◦ ¯ w (cid:96) (cid:48) νk : S (cid:96) (cid:48) νk → R × M and the maps Sh (cid:96) (cid:48) νk ◦ ¯ w : S → R × M intersect transversally at Z > C points. And hence the maps ¯ w (cid:96) (cid:48) νk : S (cid:96) (cid:48) νk → R × M and ¯ w : S → R × M intersect transversally at Z > C points. However, thiscontradicts Lemma 4.50. This is the desired contradiction which completes theproof of Theorem 7. (cid:3)
We finish Section 4.5.2 with a lemma which we use without proof.
Lemma 4.51 (stability of transversal intersections) . Let Σ and ˙Σ each be a compact manifold with boundary and diffeomorphic to theclosed two-dimensional disk { z ∈ C : | z | ≤ } . Let W be a four dimensionalmanifold, and let u : Σ → W and ˙ u : ˙Σ → W be embeddings, for which there exist ζ ∈ Σ \ ∂ Σ and ˙ ζ ∈ ˙Σ \ ∂ ˙Σ with the propertythat u ( ζ ) = ˙ u ( ˙ ζ ) and T u ( ζ ) (cid:116) T ˙ u ( ˙ ζ ) . That is, u and ˙ u intersect transversally at u ( ζ ) = ˙ u ( ˙ ζ ) . Then for any sequences { u k } k ∈ N and { ˙ u k } k ∈ N for which u k → u and ˙ u k → ˙ u in C ∞ , it is the case that for all sufficiently large k ∈ N , there exist ζ k ∈ Σ and ˙ ζ k ∈ ˙Σ with with the property that u k and ˙ u k intersect transversally at u k ( ζ k ) = ˙ u k ( ˙ ζ k ) . In addition we may assume that ζ k → ζ and ˙ ζ k → ˙ ζ . Proof of Proposition 4.49.
We begin with a restatement.
Restatement of Proposition 4.49 (feral limit curves) . Let c ≥ and let ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ¯ µ c , D c ) , be an exhaustive limit of some subsequence of the w k,c . Then ¯ w c is a feral pseudo-holomorphic curve with ¯ µ c = ∅ in the sense of Definition 1.5. As already mentioned before, the feral curve compactness theorem uses the R -action in a less systematic way then in the SFT compactness theory. As previouslyexplained the reason is that we would need a better understanding of the behaviorof the ends in a ‘generic’ situation to give a compactness theorem comparable tothe SFT-compactness theory. At this point it is not even clear what the notionof ‘generic’ has to be, or if even such a notion exists. With the current notion of ERAL CURVES AND MINIMAL SETS 143
Figure 5.
The figure shows a sequence of disks converging inan exhaustive Gromov compactness sense to a properly mappedtwo-punctured sphere. A cap flies away to ∞ creating a secondend.convergence there is in general loss of information which we shall describe by a fewexamples. As a consequence of Theorem 4, it should be clear that it is impossiblefor an arbitrarily large number of caps to “fly away to infinity” provided we have auniform ω -energy bound. Indeed, each such cap would remove at least an (cid:126) > ω -energy, leading to the absurd conclusion that the sequence of disks failed to haveuniform ω -energy bound. Figure 5 show an example of one such disk escaping toinfinity. Similarly, Figure 6 shows how it is possible for a sequence of curves withgenus one to limit to a once-punctured sphere because a handle escapes to infinity.These are just two examples of what can happen, and below we provide a compre- Figure 6.
The figure shows a sequence of disks converging inan exhaustive Gromov compactness sense to a properly mappedone-punctured sphere shedding genus.hensive discussion. We leave it to the reader to imagine an example which lackscertain uniform topology bounds (like genus, connected components, etc) and hencecan have a sequence of compact curves which develops infinitely many connectedcomponents or infinitely many nodal pairs. Indeed, without topological bounds,the limit curve can get notably wild, however a key result of Proposition 4.49 isthat the limit is a feral curve, and hence has bounded topology. The reason forthis is that producing ends or producing nodal pairs or other examples of infinitetopology either requires approximating curves to have unbounded topology or un-bounded ω -energy, each of which are excluded by the hypotheses of Proposition4.49. The proof of this result, takes some effort, which we now provide.
44 J.W. FISH AND H. HOFER
Proof.
We begin by observing that as a result of Definition 2.38 (exhaustive Gromovcompactness) and properties of the u bk , it follows that(w1) ¯ w c : S c → R × M is proper,(w2) a ◦ ¯ w c ( S c ) = [ c, ∞ ),(w3) (cid:82) S c ¯ w ∗ c ω ≤ C (w4) Genus( S c ) ≤ C .Thus, to establish that ¯ w c is feral, it remains to establish that(F1) D c < ∞ (F2) µ c = 0(F3) π ( S c ) < ∞ (F4) Punct( S c ) < ∞ where Punct( S ) is the number of generalized punctures (see Definition 1.3). Wenote that µ c = 0 since the w k,c had no marked points. Establishing the remainingproperties of a feral curve will take more effort than this, and so we first establishsome notation. For any topological space X , we will let π ( X ) denote the set ofconnected components of X , and we let π ( X ) denote the number of connectedcomponents of X . We now establish the finiteness of the number of nodal pointsand the number of connected components with Lemma 4.52 below. Lemma 4.52 (Some bounds on the limit curve) . For the pseudoholomorphic curve ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , defined above, the following inequalities hold.(1) Genus arith ( S c , ¯ j c , D c ) ≤ C (2) π ( S c ) < (cid:126) − ) C (3) D c < (cid:126) − ) C ;where Genus arith ( S c , ¯ j c , D c ) is the arithmetic genus as in Definition 2.23 and < (cid:126) = (cid:126) ( M, η, J, ¯ g, , C ) is the positive constant guaranteed by Theorem 4. We will postpone the proof of Lemma 4.52 until later; for now we continuewith the proof of Proposition 4.49, and to that end, all that remains is to estab-lish that Punct( S c ) < ∞ , which we will prove by contradiction. Thus, assumingPunct( S c ) = ∞ , we make use of the following result. Lemma 4.53 (impossible submanifold) . Let c ≥ and let ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , be an exhaustive limit of some subsequence of the w k,c . If Punct( S c ) = ∞ then there exists a compact manifold with smooth boundary Σ ⊂ S c with the followingproperties.(g1) π (Σ) = π ( S c ) (g2) π ( ∂ Σ) ≥ (cid:126) − ) C (g3) each connected component of S c \ (Σ \ ∂ Σ) is non-compact. Again we postpone the proof of Lemma 4.53 until after we have completed theproof of Proposition 4.49. We pause for a moment to collect the structure ofour argument. We are proving Proposition 4.49, which amounts to establishingproperties (F1) - (F4). Property (F2) was easily established, and properties (F1)
ERAL CURVES AND MINIMAL SETS 145 and (F3) are established by Lemma 4.52, although the proof is deferred until later.All that remains is to prove property (F4), which is that Punct( S c ) < ∞ . We willprove property (F4) by contradiction, and hence assume Punct( S c ) = ∞ , and as aconsequence of this contradiction hypothesis, we can apply Lemma 4.53, which willguarantee the existence of a compact submanifold with smooth boundary Σ ⊂ S c with a number of implausible properties. In particular, Σ will have a very largenumber of essential boundary components, and this is the feature that we willexploit in order to derive our desired contradiction, which will hence establish thatindeed Punct( S c ) < ∞ . Thus, modulo the proofs of Lemma 4.52 and Lemma 4.53,we will complete the proof of Proposition 4.49 by showing that although we have π ( ∂ Σ) ≥ (cid:126) − ) C, we must also have π ( ∂ Σ) ≤ (cid:126) − ) C ;this will be the desired contradiction.Continuing on with the proof of Proposition 4.49, we have assumed thatPunct( S ) = ∞ , and thus we may assume that the conclusions of Lemma 4.53 are true. Conse-quently, we let (cid:101) S = Σ be the surface guaranteed by Lemma 4.53, and we define( (cid:101) S, ˜ j, (cid:101) D ) to be the compact nodal Riemann surface with boundary for which ˜ j := j (cid:12)(cid:12) (cid:101) S and (cid:101) D := D ∩ (cid:101) S .Next we recall that ¯ w c is the exhaustive Gromov limit of a suitable subsequenceof the curves w k,c . We further recall that the domain of the former is ( S c , ¯ j c , D c )and the domains of the latter are ( (cid:98) S k,c , j k,c , (cid:98) D k,c ); see equation (106). Also re-call from equations (104) and (105) that we have defined the Riemann surfaces( S k,c , j k,c , D k,c ), which have the property that (cid:98) S k,c ⊂ S k,c and (cid:98) D k,c ⊂ D k,c . Wethen employ exhaustive Gromov compactness to obtain decorations (cid:101) r , ˆ r k,c , and r k,c respectively for ( (cid:101) S, ˜ j, (cid:101) D ), ( (cid:98) S k,c , j k,c , (cid:98) D k,c ) in the sense of Definition 2.22, and( S k,c , j k,c , D k,c ), and we obtain embeddings φ k : (cid:101) S (cid:101) D, ˜ r → (cid:98) S (cid:98) D k,c , ˆ r k,c k,c (cid:44) → S D k,c ,r k,c k,c , for all sufficiently large k ∈ N . We then fix some sufficiently large k ∈ N , and wedefineΣ = S D k,c ,r k,c k,c , Σ := φ k ( (cid:101) S (cid:101) D, ˜ r ) ⊂ Σ and Σ := cl (cid:0) Σ \ Σ ) . In particular, we will assume that k ∈ N has been chosen sufficiently large so thatfor each connected component Σ (cid:48) of Σ for which ∂ Σ (cid:48) ⊂ ∂ Σ we havesup ζ ∈ Σ (cid:48) a ◦ u a k + ck ( ζ ) − sup ζ ∈ ∂ Σ (cid:48) a ◦ u a k + ck ( ζ ) ≥ . That k ∈ N can be chosen sufficiently large to arrange this follows from property(g3) together with the definition of exhaustive Gromov convergence; see Definition2.38. As a consequence of this inequality, we then immediately have the following. See Theorem 2.39.
46 J.W. FISH AND H. HOFER
Lemma 4.54 (energy threshold acquired) . Let Σ (cid:48) be a connected component of Σ for which ∂ Σ (cid:48) ⊂ ∂ Σ . Then (cid:90) Σ (cid:48) ( u a k + ck ) ∗ ω ≥ (cid:126) , where (cid:126) = (cid:126) ( M, η, J, ¯ g, , C ) > is the positive constant guaranteed by Theorem 4.Proof. This follows immediately from Theorem 4. (cid:3)
The reader should note that the situation described in the lemma is the phe-nomenon where a cap, perhaps with some universally bounded genus, flies away.Each such occurrence takes at least an (cid:126) -amount of ω -energy away. If for the initialsequence of (compact) pseudoholomorphic curves the number of boundary compo-nents as well as the total ω -energy is bounded, then the number of occurrencesjust described must be bounded. Of course, we will need to establish the detailsin order to get better bounds on constants. For now we now turn our attention tomore topological estimates.We note that by property (g2) of Lemma 4.53 and the definition of Σ , we have(112) π ( ∂ Σ ) ≥ (cid:126) − ) C. We then recall that the Euler characteristic is additive, so that(113) χ (Σ ) = χ (Σ ) + χ (Σ ) . We also recall that the Euler characteristic is given by(114) χ (Σ ) = 2 π (Σ ) − ) − π ( ∂ Σ ) , and similarly for Σ and Σ . Combining equations (113) and (114), we find2 π (Σ ) − g (Σ ) − π ( ∂ Σ )= χ (Σ ) = χ (Σ ) + χ (Σ )= 2 π (Σ ) − g (Σ ) − π ( ∂ Σ )+ 2 π (Σ ) − g (Σ ) − π ( ∂ Σ )= 2 π (Σ ) − g (Σ ) − π ( ∂ Σ )+ 2 π (Σ ) − g (Σ ) − (cid:0) π ( ∂ Σ ) + π ( ∂ Σ ) (cid:1) , where we have simplified the notation by writing g (Σ ) = Genus(Σ ), and to obtainthe final equality, we have made use of the following observation: π ( ∂ Σ ) = π ( ∂ Σ ) + π ( ∂ Σ ) . After rearranging and simplifying, we have the following estimate. π ( ∂ Σ ) = g (Σ ) − g (Σ ) − g (Σ ) + π (Σ ) + π (Σ ) − π (Σ ) ≤ g (Σ ) + π (Σ ) + π (Σ )(115)Next we estimate the genus of Σ , as follows:Genus(Σ ) = Genus( S D k,c ,r k,c k,c )= Genus arith ( S k,c , j k,c , D k,c )= π ( | S k,c | ) − π ( S k,c ) + Genus( S k,c ) + D k,c ≤ π ( | S k,c | ) + Genus( S k,c ) + D k,c ≤ C, (116) ERAL CURVES AND MINIMAL SETS 147 where to obtain the third equality we have employed Lemma A.1 from [13], and toobtain the final inequality, we have used the fact that S k,c = S a k + ck , and hence bythe assumptions of Theorem 7 we have π ( | S k,c | ) = 1 ≤ C , Genus( S k,c ) ≤ C , and D k,c ≤ C ≤ C . Combining inequality (115) with inequality (116) then yields π ( ∂ Σ ) ≤ C + π (Σ ) + π (Σ ) . (117)We can then estimate π (Σ ) as follows. π (Σ ) = π ( (cid:101) S (cid:101) D, ˜ r ) by Definition of Σ ≤ π ( (cid:101) S ) by properties of nodal curves= π ( S c ) by Lemma 4.53 ≤ (cid:126) − ) C by Lemma 4.52Or in other words,(118) π (Σ ) ≤ (cid:126) − ) C. To proceed further, we partition Σ into three disjoint sets denoted Σ bdry , Σ int ,and Σ const ; here Σ const consists of connected components of Σ on which the map u k is constant, Σ bdry consists of connected components of Σ which have non-trivialintersection with ∂ Σ , and we define Σ int := Σ \ (Σ const ∪ Σ bdry ).As a consequence of the fact that the number of connected components of the ∂S bk is uniformly bounded by C , it follows from the definition of Σ and Σ bdry thatwe must have π (Σ bdry ) ≤ C . Also, because the curves u bk = (cid:0) u bk , S bk , j bk , ( −∞ , × M, J k , ∅ , D bk (cid:1) are stable and without marked points, it follows that each connected component ofΣ const must contain a nodal point in D bk . Recalling that D bk ≤ C it follows that π (Σ const ) ≤ C . Combining these two inequalities then yields(119) π (Σ const ) + π (Σ bdry ) ≤ C. Lastly we note that Σ int consists of connected components on which u bk is non-constant, and ∂ Σ int ⊂ ∂ Σ , and hence by Lemma 4.54 and the assumption thatour curves have ω -energy bounded by C , we have(120) (cid:126) · π (Σ int ) ≤ (cid:90) Σ int u ∗ k,c ω ≤ C. Here we have abused notation somewhat since, strictly speaking, Σ int is a circle-compactified surface rather than a domain of a pseudoholomorphic curve, howeverthis can be made rigorous by removing the added-special circles from Σ int in theabove integral; in any case, the desired estimate (cid:126) · π (Σ int ) ≤ C holds. Combininginequalities (118), (119), and (120) with inequality (117) then yields π ( ∂ Σ ) ≤ C + 6(1 + (cid:126) − ) C + 2 C + (cid:126) − C ≤ (cid:126) − ) C. However, combining the above inequality with inequality (112)12(1 + (cid:126) − ) C ≤ π ( ∂ Σ ) ≤ (cid:126) − ) C
48 J.W. FISH AND H. HOFER which is the desired contradiction, which establishes that we must have Punct( S c ) < ∞ . Thus, modulo the proofs of Lemma 4.52 and Lemma 4.53, we have completedthe proof of Proposition 4.49. (cid:3) We now turn our attention to the proof of Lemma 4.52. We begin with a re-statement.
Restatement of Lemma 4.52 (Some bounds on the limit curve) . For the pseudoholomorphic curve ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , defined above, the following inequalities hold.(1) Genus arith ( S c , ¯ j c , D c ) ≤ C (2) π ( S c ) < (cid:126) − ) C (3) D c < (cid:126) − ) C ;where Genus arith ( S c , ¯ j c , D c ) is the arithmetic genus as in Definition 2.23 and < (cid:126) = (cid:126) ( M, η, J, ¯ g, , C ) is the positive constant guaranteed by Theorem 4.Proof. In an effort to simplify notation a bit, we will drop the subscripts c , andwrite, for example, ¯ w and S instead of ¯ w c and S c .We begin by recalling Definition 2.23 which guarantees thatGenus arith ( S, j, µ, D ) = Genus( S D,r ) . Moreover, the genus of a non-compact surface is obtained as the limit of generaof an exhausting sequence of compact surfaces with boundary. By genus super-additivity , the definition of exhaustive Gromov compactness , and properties ofthe S bk , it follows thatGenus arith ( S, ¯ j, D ) ≤ sup k,b Genus arith ( S bk , j bk , D bk )(121)However, recall Lemma A.1 from [13] which provides a formula for the arithmeticgenus of a compact Riemann surface with boundary:Genus arith ( S, j, D )(122) = π ( | S | ) − π ( S ) + (cid:16) (cid:88) Σ ∈ π ( S ) Genus(Σ) (cid:17) + D. In light of the bounds we have on Genus( S bk ) and D bk due to the hypotheses ofTheorem 7, we immediately see that(123) Genus arith ( S, ¯ j, D ) ≤ C ≤ C. This establishes the first desired inequality; the next two will require a bit moreeffort.We pause for a moment to highlight the difficulty in proving the second desiredinequality, namely that π ( S ) < (cid:126) − ) C. See Lemma 2.29. See Definition 2.38.
ERAL CURVES AND MINIMAL SETS 149
If the map ¯ w were non-constant on each connected component of S , then of coursethe estimate (in fact a better estimate) would follow quickly. Thus the main diffi-culty is to establish a bound on the number of constant components. Because ¯ w is stable, we could easily bound the number of constant components in terms ofthe number of nodal points, but we do not have an a priori bound on that either,since the number of nodal points can increase in the exhaustive Gromov limit of asequence of curves. Finally, it would also be easier to bound the number of constantcomponents if we knew that either the non-constant components were compact orwe knew that the number of nodal points was finite, however a priori we knowneither of these. As such, the path to obtaining the desired bound may not seemstraightforward, even though the basic idea is; that is, we essentially aim to usethe stability condition plus an energy threshold to bound the number of constantcomponents in terms of ω -energy. This is the tack we take, and we return to theproof presently.The next step is to define the set I reg ⊂ R to be the intersection of the set[ c, ∞ ) \ a ◦ ¯ w ( D ) ⊂ ( c, ∞ ) with the set of regular values of the function a ◦ ¯ w : S → R .We note that I reg is an open and dense subset of ( c, ∞ ). Next, for each x ∈ I reg we define a compact nodal Riemann surface ( S x , j x , D x ) in the following manner.First, we enumerate the connected components of S by S k , so that S = (cid:83) ∞ k =1 S k .Next, on each connected component S k we choose ζ k ∈ S k so thatinf ζ ∈ S k a ◦ ¯ w ( ζ ) = a ◦ ¯ w ( ζ k ) . We denote the collection of these points by Z = { ζ , ζ , . . . } . For each x ∈ I reg wethen define Σ x := ( a ◦ ¯ w ) − (cid:0) ( −∞ , x ] (cid:1) S x := (cid:8) Σ ∈ π (Σ x ) : Z ∩ Σ (cid:54) = ∅ and a ◦ ¯ w ( Z ∩ Σ) ≤ x − (cid:9) Observe that S x is a finite set. It is worth pausing to describe this set S x . Indeed,this can be regarded as a set of “essential” connected components of Σ x , where byessential we mean those components which contain both a marker ζ k which identifiesconnected components of S , and those components on which the minimum valueof a ◦ ¯ w differs from x (which will often be that maximal value of a ◦ ¯ w ) by atleast 1. We will exploit these features momentarily, but we first must continue ourdefinition of S x .Next we aim to define a certain collection of Riemann surfaces which we denoteStab x . To do this, we let 2 S x denote the power set of S x , we let σ : D → D denotethe involution satisfying σ ( d i ) = d i and σ ( d i ) = d i for each d i , d i ∈ D and we saya triple ( (cid:101) Σ , ˜ j, (cid:101) D ) is ¯ w -stable provided it is a nodal Riemann surface with (cid:101) Σ ⊂ S (cid:101) D ⊂ D , and for each connected component Σ ⊂ (cid:101) Σ for which ¯ w : Σ → W is constantwe have 2Genus(Σ) + (cid:101) D ∩ Σ) ≥ . We then define Stab x via the following.Stab x := (cid:110) ( (cid:101) Σ , ˜ j, (cid:101) D ) : (cid:101) Σ = (cid:91) Σ ∈A Σ where
A ∈ S x , ˜ j = j (cid:12)(cid:12) (cid:101) Σ (cid:101) D ⊂ (cid:101) Σ ∩ D satisfies σ ( (cid:101) D ) = (cid:101) D and ( (cid:101) Σ , ˜ j, (cid:101) D ) is w -stable (cid:111)
50 J.W. FISH AND H. HOFER ζ ζ ζ ‘ “ S x ζ ζ S x ‘ Figure 7.
Two examples of the set S x . In the left figure it has twoelements and in the right figure one element. These componentsare obtained from the sets indicated by taking those points forwhich a ◦ ¯ w takes a value not exceeding x .Observe that S x for x ∈ ( c, ∞ ) is nonempty. We introduce a partial order on Stab x by defining ( (cid:101) Σ , ˜ j , (cid:101) D ) ≤ ( (cid:101) Σ , ˜ j , (cid:101) D ) if and only if (cid:101) Σ ⊂ (cid:101) Σ and (cid:101) D ⊂ (cid:101) D . Finally,we note that given two elements ( (cid:101) Σ , ˜ j , (cid:101) D ) , ( (cid:101) Σ , ˜ j , (cid:101) D ) ∈ Stab x their union (inthe obvious manner) is again in Stab x , and hence the partially ordered set Stab x has a greatest element. We define ( S x , j x , D x ) to be the greatest element of Stab x .At this point, we have defined the compact nodal Riemann surface ( S x , j x , D x ),which may have boundary.The definition provided may seem convoluted, however it has a number of fea-tures we now state and will exploit momentarily.First, we note that for each x, y ∈ I reg with x < y , we have π ( S x ) ≤ π ( S y ) ≤ π ( S ), and (cid:83) x ∈I reg S x = S , from which we conclude that(124) lim x →∞ π ( S x ) = π ( S ) . Similarly, for each x, y ∈ I reg with x < y , we haveGenus arith ( S x , j x , D x ) ≤ Genus arith ( S y , j y , D y ) , and by definition of the arithmetic genus, we have(125) lim x →∞ Genus arith ( S x , j x , D x ) = Genus arith ( S, ¯ j, D ) . Second, we let (cid:126) = (cid:126) ( M, η, J, ¯ g > , , C ) > S x on which ¯ w is non-constant, we have (cid:90) Σ ¯ w ∗ ω ≥ (cid:126) . To make use of this property, we first decompose S x into two sets, S xconst and S xnc ,where S xconst is the union of connected components on which ¯ w is constant and ERAL CURVES AND MINIMAL SETS 151 S xnc = S x \ S xconst , and we then recall that ω evaluates non-negatively on J -complexlines, so that by properties of the w k and exhaustive Gromov compactness, we have(126) C ≥ (cid:90) S xnc ¯ w ∗ ω ≥ (cid:126) · π ( S xnc ) . Third, the ¯ w -stability condition guarantees that for each Σ ∈ S xconst we have2Genus(Σ) + D x ∩ Σ) ≥ . To make use of this, it will be convenient to define S xconst ( k ) := (cid:110) Σ x ∈ π ( S xconst ) : Genus(Σ x ) = k (cid:111) , in which case we can estimate:(127) 3 π (cid:0) S xconst (0) (cid:1) + π (cid:0) S xconst (1) (cid:1) ≤ D x . We are now prepared to complete the proof of Lemma 4.52. As above, we have aformula for the arithmetic genus of ( S x , j x , D x ) given byGenus arith ( S x , j x , D x )(128) = π ( | S x | ) − π ( S x ) + (cid:16) (cid:88) Σ x ∈ π ( S x ) Genus(Σ x ) (cid:17) + D x . We then note that π ( S x ) = π ( S xnc ) + ∞ (cid:88) k =0 π (cid:0) S xconst ( k ) (cid:1) , and we recall that (cid:88) Σ x ∈ π ( S x ) Genus(Σ x ) = Genus( S xnc )+ ∞ (cid:88) k =1 k · π (cid:0) S xconst ( k ) (cid:1) ≥ ∞ (cid:88) k =1 k · π (cid:0) S xconst ( k ) (cid:1) which is finite since sup { k ∈ N : S xconst ( k ) (cid:54) = ∅} ≤ Genus( S ) ≤ C .Combining the above two (in)equalities with inequality (127) and the formulafor the arithmetic genus then yields the following.Genus arith ( S x , j x , D x )= π ( | S x | ) − π ( S x ) + (cid:16) (cid:88) Σ x ∈ π ( S x ) Genus(Σ x ) (cid:17) + D x ≥ π ( | S x | ) − π ( S xnc ) − ∞ (cid:88) k =0 π (cid:0) S xconst ( k ) (cid:1) + ∞ (cid:88) k =1 k · π (cid:0) S xconst ( k ) (cid:1) + π (cid:0) S xconst (0) (cid:1) + π (cid:0) S xconst (1) (cid:1) = π ( | S x | ) − π ( S xnc ) + ∞ (cid:88) k =1 ( k − · π (cid:0) S xconst ( k ) (cid:1) + π (cid:0) S xconst (0) (cid:1) + π (cid:0) S xconst (1) (cid:1) ≥ π ( | S x | ) − π ( S xnc ) + π ( S xconst ) ≥ − π ( S xnc ) + π ( S xconst ) .
52 J.W. FISH AND H. HOFER
Or in other words,2Genus arith ( S x , j x , D x ) + 2 π ( S xnc ) ≥ π ( S xconst ) , and thus π ( S x ) ≤ arith ( S x , j x , D x ) + 3 π ( S xnc ) . Next, we recall equations (124), (125) and (126), which guarantee the following π ( S ) = lim x →∞ π ( S x ) ≤ lim x →∞ arith ( S x , j x , D x ) + 3 (cid:126) − C = 2Genus arith ( S, ¯ j, D ) + 3 (cid:126) − C ≤ C ) + (cid:126) − C, ≤ (cid:126) − ) C where to obtain the second inequality we have employed inequality (123). Thisestablishes the desired bound on the number of connected components of S , andproves the second part of the conclusions of Lemma 4.52.To establish the third part of Lemma 4.52, we recall equation (128), which statesthe following.Genus arith ( S x , j x , D x )= π ( | S x | ) − π ( S x ) + (cid:16) (cid:88) Σ x ∈ π ( S x ) Genus(Σ x ) (cid:17) + D x Solve for D x and pass to the limit as x → ∞ in I reg to obtain the following: D = 2Genus arith ( S, ¯ j, D ) − π ( | S | ) + 2 π ( S ) − (cid:16) (cid:88) Σ ∈ π ( S ) Genus(Σ) (cid:17) ≤ arith ( S, ¯ j, D ) + 2 π ( S ) ≤ C + 12(1 + (cid:126) − ) C ≤ (cid:126) − ) C This is the desired estimate, which then completes the proof of Lemma 4.52. (cid:3)
At this point we note that we have proved Proposition 4.49 modulo only theproof of Lemma 4.53, and so we turn our attention to that. First however, it willbe important to define a procedure called a cut . We make the definition precisebelow.
Definition 4.55 (cut) . Let u : S → R × M be a proper pseudoholomorphic map. Letting I reg denote theregular values of a ◦ u , and assuming a ◦ u ( z ) ∈ I reg , we define cut z ( S ) by firstdefining Γ z to be the connected component of ( a ◦ u ) − (cid:0) a ◦ u ( z ) (cid:1) containing z , andwe let cut z ( S ) be the surface obtained by gluing in two disjoint copies of Γ z into S \ Γ z . For example, suppose u : R × S → R × M is a proper pseudoholomorphiccylinder for which the function a ◦ u has no critical points, then for each z ∈ R × S the surface cut z ( R × S ) is diffeomorphic to the disjoint union of ( −∞ , × S and[0 , ∞ ) × S .We now recall what we will prove. ERAL CURVES AND MINIMAL SETS 153
Restatement of Lemma 4.53 (impossible submanifold) . Let c ≥ and let ¯ w c = ( ¯ w c , S c , ¯ j c , R × M, J, ∅ , D c ) , be an exhaustive limit of some subsequence of the w k,c . If Punct( S c ) = ∞ then there exists a compact manifold with smooth boundary Σ ⊂ S c with the followingproperties.(g1) π (Σ) = π ( S c ) (g2) π ( ∂ Σ) ≥ (cid:126) − ) C (g3) each connected component of S c \ (Σ \ ∂ Σ) is non-compact.Proof. As in the proof of Lemma 4.52, we will attempt to simplify notation a bitby dropping the subscripts c , and writing, for example, ¯ w and S instead of ¯ w c and S c .Our first step is to put precisely one special point, ζ k , on each connected compo-nent of S . We denote the set of such points Z = { ζ , . . . , ζ n } , and note that this setis finite as a consequence of Lemma 4.52. By assumption we have Punct( ¯ w ) = ∞ ,so it follows that there exists x > x > x , thenumber of non-compact connected components of S \ ( a ◦ ¯ w ) − (( −∞ , x )) is greaterthan or equal to 12(1 + (cid:126) − ) C . To make use of this, we first define I reg to be theintersection of the sets R \ a ◦ ¯ w ( D ) and the set of regular values of the function a ◦ ¯ w : S → R . We then choose x sufficiently large so that for each x ∈ I reg with x > x , and for Σ x := ( a ◦ ¯ w ) − (( −∞ , x ]) we have(1) Z ∪ D ⊂ Σ x ,(2) Genus(Σ x ) = Genus( S ),(3) each compact connected component of S is contained in Σ x ,(4) the number of non-compact connected components of S \ (Σ x \ ∂ Σ x ) isgreater than 12(1 + (cid:126) − ) C .We note that the existence of such a x relies both on the validity of Lemma 4.52 andthe assumption that Punct( ¯ w ) = ∞ . We henceforth assume x ∈ I reg with x > x has been fixed. We also note an important property, namely that as a consequenceof the fact that Genus(Σ x ) = Genus( S ), it follows that any embedded loop removedfrom S \ Σ x disconnects the surface S ; this follows from genus super-additivity andan straightforward Euler characteristic argument.Next, we enumerate the set of non-compact connected components of S \ (Σ x \ ∂ Σ x ) as E , . . . , E m with m ≥ (cid:126) − ) C. Also, for each k ∈ { , . . . , m } we choose a continuous path γ k : [0 , → S, eachwith the property that γ k (0) ∈ Z and γ k (1) ∈ ∂E k . At this point, we fix a x (cid:48) ∈ I reg with x (cid:48) > x with the additional property that m (cid:91) k =1 γ k (cid:0) [0 , (cid:1) ⊂ Σ x (cid:48) . Next, we enumerate the non-compact ends of S \ (Σ x (cid:48) \ ∂ Σ x (cid:48) (cid:1) via E (cid:48) , E (cid:48) , . . . , E (cid:48) m (cid:48) .We also extend each γ k : [0 , → S to continuous γ k : [0 , → S so that(129) γ k (1 , ⊂ S \ Σ x
54 J.W. FISH AND H. HOFER xZ x’
Figure 8.
The figure illustrates the construction. It shows x , x (cid:48) and the extended curves γ k . The actual situation can be ingenerally much wilder. In our case we have above x (cid:48) only non-compact components (not shown), i.e. E (cid:48) ,..., E (cid:48) m . A later figurewill show additional possible features.and for each k ∈ { , . . . , m } we have(130) γ k ( t ) ∈ ∪ m (cid:48) i =1 E (cid:48) i if and only if t = 2 . We then obtain a new surface, denoted by (cid:101)
Σ, by cutting S at the circles associatedto the points γ (2) , . . . , γ m (2), and defining (cid:101) Σ to be the union of the connectedcomponents of the cut surface which have non-empty intersection with Z .We pause for a moment to consider the properties of the surface (cid:101) Σ, since it isclose to the surface we seek. To that end, we first observe that ∂ (cid:101) Σ ⊂ ∪ m (cid:48) k =1 ∂E (cid:48) k ;this follows from equations (129) and (130).Second, we claim that π ( (cid:101) Σ) = n = π ( S ). To see this, first note thatby definition Z ⊂ (cid:101) Σ ⊂ S , and each element of Z lies on a different connectedcomponent of S , and hence π ( (cid:101) Σ) ≥ π ( S ) = n ; and because each connectedcomponent of (cid:101) Σ must contain a point in Z , the opposite inequality must hold aswell.Third, we claim that π ( ∂ (cid:101) Σ) = m ≥ (cid:126) − ) C . To establish this, it isimportant to observe that ∪ mk =1 γ k ([0 , ⊂ (cid:101) Σ. To see this, recall that equation(130) guarantees that for each k ∈ { , . . . , m } , we have γ k ( t ) ∈ ∪ m (cid:48) i =1 E i if andonly if t = 2; furthermore, since ∂ (cid:101) Σ ⊂ ∪ m (cid:48) k =1 ∂E (cid:48) k , it follows that each γ k ([0 , S atthe circles associated to the points γ (2) , . . . , γ m (2). However, because γ k (0) ∈ Z for k ∈ { , . . . , m } , and Z ⊂ (cid:101) Σ, it follows that indeed, ∪ mk =1 γ k ([0 , ⊂ (cid:101) Σ. Wecan now prove that π ( ∂ (cid:101) Σ) = m . To see this, recall that by construction, foreach k ∈ { , . . . , m } , the point γ k (1) is an element of a connected component of ERAL CURVES AND MINIMAL SETS 155 xx’ xx’ xx’x’’
Figure 9.
Left: In this case Z consists of one point and wehave four non-compact components E , .., E . We also have four E (cid:48) , ., E (cid:48) . Right: The set (cid:101) Σ, which we note is not compact. Thisset has already a lot of desirable properties. Below: The desiredset Σ is obtained by trimming it further. S \ (Σ x \ ∂ Σ x ), and moreover no two such points γ k (1) and γ k (cid:48) (1) are contained inthe same connected component of S \ (Σ x \ ∂ Σ x ). Furthermore, by equation (129)we have γ k ((1 , ⊂ S \ Σ x , and since γ k (2) ∈ ∂ (cid:101) Σ for each k ∈ { , . . . , m } , it followsthat π ( ∂ (cid:101) Σ) ≥ m . The equality π ( ∂ (cid:101) Σ) = m follows from the fact that eachconnected component of ∂ (cid:101) Σ contains one point of the form γ k (2).Fourth, and finally, we claim that each connected component of S \ ( (cid:101) Σ \ ∂ (cid:101) Σ) is non-compact. To establish this, let us define (cid:98)
Σ to be the surface obtained by cutting S atthe circles associated to the points γ (2) , . . . , γ m (2); recall that (cid:101) Σ is then defined tobe the union of the connected components of (cid:98)
Σ which have non-empty intersectionwith Z . Consequently, suppose Σ is a compact connected component of (cid:98) Σ . Thereare two cases to consider. In the first case, ∂ Σ = ∅ , in which case it follows thatΣ ∩ Z (cid:54) = ∅ by definition of Z . In the second case, ∂ Σ (cid:54) = ∅ , it follows that ∂ Σ has non-trivial intersection with the set { γ (2) , γ (2) , . . . , γ m (2) } . It then follows from thefact that ∂ (cid:101) Σ ⊂ ∪ m (cid:48) k =1 ∂E (cid:48) k , that either Σ ⊂ (cid:101) Σ or else Σ ⊂ ∪ m (cid:48) k =1 E (cid:48) k . However, since
56 J.W. FISH AND H. HOFER each connected component of ∪ m (cid:48) k =1 E (cid:48) k is non-compact and Σ is compact, it followsthat we must have Σ ⊂ (cid:101) Σ. Thus whenever Σ is a compact connected componentof (cid:98)
Σ, we have Σ ⊂ (cid:101) Σ . Summarizing, we have constructed a surface (cid:101) Σ ⊂ S with theproperties(1) ∂ (cid:101) Σ ⊂ ∪ m (cid:48) k =1 ∂E (cid:48) k (2) π ( (cid:101) Σ) = n = π ( S )(3) π ( ∂ (cid:101) Σ) = m ≥ (cid:126) − ) C .(4) each connected component of S \ ( (cid:101) Σ \ ∂ (cid:101) Σ) is non-compact.Observe that we would have found the desired Riemann surface if only (cid:101)
Σ hadbeen compact. Since (cid:101)
Σ need not be compact, we must trim it further to obtainthe desired surface. To that end, we fix, x (cid:48)(cid:48) ∈ I reg with x (cid:48)(cid:48) > x (cid:48) . We then define E (cid:48)(cid:48) to be the union of the interiors of the non-compact connected components of (cid:101) Σ ∩ ( a ◦ w ) − ([ x (cid:48)(cid:48) , ∞ )), and we define Σ to be the set of all points in p ∈ (cid:101) Σ forwhich there exists a continuous path in (cid:101) Σ \ E (cid:48)(cid:48) from x to Z .We now establish the required properties. First we note that (cid:101) Σ \ E (cid:48)(cid:48) is compact,and Σ ⊂ (cid:101) Σ \ E (cid:48)(cid:48) is closed, so that Σ is indeed compact. Again, every connectedcomponent of Σ is path-connected to Z , and hence π (Σ) = n = π ( S ). Alsoby construction ∂ (cid:101) Σ ⊂ ∂ Σ, and hence π ( ∂ Σ) ≥ π ( ∂ (cid:101) Σ) = m ≥ (cid:126) − ) C. Finally, we claim that each connected component of S \ (Σ \ ∂ Σ) is non-compact. Tosee this, we first note that each connected component of S \ ( (cid:101) Σ \ ∂ (cid:101) Σ) is a connectedcomponent of S \ (Σ \ ∂ Σ), and we have already established that each of these is non-compact. Thus it is sufficient to show that the connected components of (cid:101) Σ \ (Σ \ ∂ Σ)are non-compact. Observe that any connected component of (cid:101) Σ \ (Σ \ ∂ Σ) havingnontrivial intersection with E (cid:48)(cid:48) must be non-compact. However, by definition ofΣ, any point p ∈ (cid:101) Σ \ (Σ \ ∂ Σ) has the property that every path connecting p to Z will intersect E (cid:48)(cid:48) . In other words, the connected component of (cid:101) Σ \ (Σ \ ∂ Σ)which contains such a p must also contain a connected component of E (cid:48)(cid:48) , and hencemust be non-compact. This establishes all the required properties of Σ, and hencecompletes the proof of Lemma 4.53. (cid:3) We now observe that we have completed the proof of Proposition 4.49, includingall dependencies.4.5.4.
Proof of Lemma 4.50.
Restatement of Lemma 4.50 (bounded transverse intersections) . Consider non-negative numbers c, c (cid:48) ≥ with c (cid:48) > c , and let k (cid:55)→ (cid:96) k ∈ N be a strictlyincreasing sequence for which w (cid:96) k ,c → ¯ w c and w (cid:96) k ,c (cid:48) → ¯ w (cid:48) c in an exhaustive sense.Then the subset P ⊂ R × M of transversal intersection points of the two curves,which is defined by P := (cid:8) p ∈ R × M : there exists ( ζ, ζ (cid:48) ) ∈ S c × S c (cid:48) such that ¯ w c ( ζ ) = p = ¯ w c (cid:48) ( ζ (cid:48) ) and T ¯ w c ( ζ ) (cid:116) T ¯ w c (cid:48) ( ζ (cid:48) ) (cid:9) , satisfies P ≤ C. ERAL CURVES AND MINIMAL SETS 157
Proof.
We will proceed via a proof by contradiction, and assume that P > C .Consequently there exist distinct p , . . . , p n ∈ R × M with n > C , and there exist ζ , . . . , ζ n ∈ S and ζ (cid:48) , . . . , ζ (cid:48) n ∈ S (cid:48) with the property that¯ w c ( ζ k ) = p k = ¯ w c (cid:48) ( ζ (cid:48) k ) and T ¯ w c ( ζ k ) (cid:116) T ¯ w c (cid:48) ( ζ (cid:48) k )for each k ∈ { , . . . , n } . Because the p , . . . , p n are distinct and because ¯ w c and¯ w c (cid:48) are respectively immersions at the points ζ k and ζ (cid:48) k for each k ∈ { , . . . , n } itfollows that we may find closed disks ∆ , . . . , ∆ n ⊂ S c and ∆ (cid:48) , . . . , ∆ (cid:48) n ⊂ S c (cid:48) whichare pairwise disjoint, and satisfy ζ k ∈ ∆ k \ ∂ ∆ k and ζ (cid:48) k ∈ ∆ (cid:48) k \ ∂ ∆ (cid:48) k , and for whichthe maps ¯ w c : n (cid:91) k =1 ∆ k → R × M and ¯ w c (cid:48) : n (cid:91) k =1 ∆ (cid:48) k → R × M are embeddings.We then note as a consequence of the definition of exhaustive Gromov compact-ness, there exist, for all sufficiently large k ∈ N , embeddings φ (cid:96) k : n (cid:91) ν =1 ∆ ν → S (cid:96) k ,c and φ (cid:48) (cid:96) k : n (cid:91) ν =1 ∆ (cid:48) ν → S (cid:96) k ,c (cid:48) with the property that the maps w (cid:96) k ,c ◦ φ (cid:96) k : n (cid:91) ν =1 ∆ ν → R × M and w (cid:96) k ,c (cid:48) ◦ φ (cid:48) (cid:96) k : n (cid:91) ν =1 ∆ (cid:48) ν → R × M respectively converge in C ∞ to¯ w c : n (cid:91) ν =1 ∆ ν → R × M and ¯ w c (cid:48) : n (cid:91) ν =1 ∆ (cid:48) ν → R × M. Then by Lemma 4.51, it follows that for all sufficiently large k ∈ N there existdistinct z , . . . , z n ∈ φ (cid:96) k (cid:16) n (cid:91) ν =1 ∆ ν (cid:17) ⊂ S (cid:96) k ,c and z (cid:48) , . . . , z (cid:48) n ∈ φ (cid:48) (cid:96) k (cid:16) n (cid:91) ν =1 ∆ (cid:48) ν (cid:17) ⊂ S (cid:96) k ,c (cid:48) for which w (cid:96) k ,c ( z ν ) = w (cid:96) k ,c (cid:48) ( z (cid:48) ν ) for ν ∈ { , . . . , n } . Recall equation (107) whichguarantees w k,c = Sh a k ◦ u a k + ck where Sh x is the shift map Sh x ( a, p ) = ( a − x, p ), and the u a k + ck are of the curves u bk specified in the hypotheses of Theorem 7. Consequently,Sh a (cid:96)k ◦ u a (cid:96)k + c(cid:96) k ( z ν ) = w (cid:96) k ,c ( z ν ) = w (cid:96) k ,c (cid:48) ( z (cid:48) ν ) = Sh a (cid:96)k ◦ u a (cid:96)k + c (cid:48) (cid:96) k ( z (cid:48) ν )and hence u a (cid:96)k + c(cid:96) k ( z ν ) = u a (cid:96)k + c (cid:48) (cid:96) k ( z (cid:48) ν )for ν ∈ { , . . . , n } with n > C . Recall equation (104) which guarantees that S (cid:96) k ,c = S a (cid:96)k + c(cid:96) k and S (cid:96) k ,c (cid:48) = S a (cid:96)k + c (cid:48) (cid:96) k , and hence we conclude that (cid:0) u a (cid:96)k + c(cid:96) k ( S a (cid:96)k + c(cid:96) k ) ∩ u a (cid:96)k + c (cid:48) (cid:96) k ( S a (cid:96)k + c (cid:48) (cid:96) k ) (cid:1) > C,
58 J.W. FISH AND H. HOFER for all sufficiently large k ∈ N . However this contradicts the hypothesis of Theorem7 which states (cid:0) u bk ( S bk ) ∩ u b (cid:48) k ( S b (cid:48) k ) (cid:1) ≤ C for all b, b (cid:48) ≥ k ∈ N . This is the contradiction we have sought, and hence theproof of Lemma 4.50. (cid:3) Appendix A. Minor Miscellanea
A.1.
Riemannian Recollections.
Let (
M, g ) be a Riemannian manifold of di-mension n . Recall that the metric g uniquely determines a torsion-free metricconnection, called the Levi-Civta connection. We denote the associated covariantderivative by ∇ . That is, ∇ satisfies ∇ X Y − ∇ Y X = [ X, Y ] and ∇(cid:104)
X, Y (cid:105) g = (cid:104)∇ X, Y (cid:105) g + (cid:104) X, ∇ Y (cid:105) g . Let p ∈ M , and let x = ( x , . . . , x n ) be local coordinates near p so that x ( p ) = 0 ∈ R n . We express g in local coordinates by the following. g = g ij dx i ⊗ dx j Here, and throughout, we employ Einstein’s notation for summing over repeatedindices. We also uniquely define n functions g ij by the equations g i(cid:96) g (cid:96)j = δ ij with δ ij the Kronecker delta. In this case we may express ∇ in local coordinates as ∇ X i ∂ xi ( Y j ∂ x j ) = X i dY j ( ∂ x i ) ∂ x j + X i Y j Γ kij ∂ x k , where Γ kij are the Christoffel symbols, which are given by(131) Γ kij = g k(cid:96) (cid:0) g i(cid:96),j + g j(cid:96),i − g ij,(cid:96) (cid:1) where g ij,k = ∂∂x k g ij . It is worth noting that0 = ∇ ∂ xk ( δ ij )= ∇ ∂ xk ( dx i ( ∂ x j ))= ( ∇ ∂ xk dx i )( ∂ x j ) + dx i ( ∇ ∂ xk ∂ x j )= ( ∇ ∂ xk dx i )( ∂ x j ) + dx i (Γ (cid:96)kj ∂ x (cid:96) )= ( ∇ ∂ xk dx i )( ∂ x j ) + Γ ikj from which we conclude that ∇ ∂ xk dx i = − Γ (cid:96)ki dx (cid:96) . Recall that given a point p ∈ M and a (sufficiently small) vector Z ∈ T p M ,there exists unique geodesic emanating from p with initial velocity Z . That is tosay, there exists a unique solution γ : [0 , → M to the initial value problem(132) ∇ ˙ γ ( t ) ˙ γ ( t ) = 0 and γ (0) = p, γ (cid:48) (0) = Z. The exponential map associated to g , denoted exp gp : T p M → M , is defined byexp gp ( Z ) = γ (1) where γ solves the differential equation (132). Furthermore, givenan orthonormal basis ( Z , . . . , Z n ) of T p M , we define normal geodesic polar coor-dinates x = ( x , . . . , x n ) near p by the following x i ( q ) = (cid:10) (exp gp ) − ( q ) , Z i (cid:11) g . ERAL CURVES AND MINIMAL SETS 159
Recall that in these normal geodesic coordinates, we have the following(133) g ij ( p ) = δ ij and Γ kij ( p ) = 0 . Furthermore, we also have(134) ∂∂x k g ij ( p ) = 0 . To see that equation (134) holds, we simply compute ∂∂x k g ij = ∇ ∂ xk (cid:104) ∂ x i , ∂ x j (cid:105) g = (cid:104)∇ ∂ xk ∂ x i , ∂ x j (cid:105) g + (cid:104) ∂ x i , ∇ ∂ xk ∂ x j (cid:105) g = (cid:104) Γ (cid:96)ki ∂ x (cid:96) , ∂ x j (cid:105) g + (cid:104) ∂ x i , Γ (cid:96)kj ∂ x (cid:96) (cid:105) g evaluating at p and employing equation (133) then establishes equation (134).We now consider R × M with ( M, g ) as above, a coordinate a on R , and themetric ¯ g := da ⊗ da + g . Fix a point ( p , p ) ∈ R × M and fix ( Z , Z , . . . , Z n ) anorthonormal basis of T p M , and let ( x , . . . , x n ) denote the associated normal geo-desic coordinates defined near p . We extend these to coordinates ( x , x , . . . , x n )by taking x = a . We now claim the following. Lemma A.1 (properties of g and Γ) . In the coordinates ( x , . . . , x n ) established above, the following hold. (135) ¯ g ij ( p , p ) = δ ij and Γ kij ( p , p ) = 0 , where Γ kij are the Christoffel symbols associated to ¯ g in the coordinates ( x , . . . , x n ) .Moreover, in these local coordinates, we have (136) ∂∂x ¯ g ij = 0 = ∂∂x Γ kij . Proof.
For ease of notation, we write ¯ p = ( p , p ). Begin by observing that whenever0 / ∈ { i, j, k } we have ¯ g ij (¯ p ) = δ ij and Γ kij (¯ p ) = 0. This follows from the fact thatthe ( x , . . . , x n ) for normal geodesic coordinates associated to g . Next observe thatby definition of ¯ g we have ¯ g = 1, and because ¯ g is a product metric, it followsthat ¯ g j = 0 = ¯ g j for j ∈ { , . . . , n } . These results establish the first equality inequation (135).We next aim to prove that Γ kij (¯ p ) = 0 when 0 ∈ { i, j, k } . To that end, first observethat ¯ g i = ¯ g i = δ i , ¯ g ij = g ij whenever 0 / ∈ { i, j } , and all ¯ g ij are independent of x . This latter fact together with the formula for the Christoffel symbols givenin equation (131) then guarantee equation (136). Furthermore, the term ( g j(cid:96),i + g i(cid:96),j − g ij,(cid:96) ) in equation (131) vanishes whenever 0 ∈ { i, j, (cid:96) } . If 0 / ∈ { i, j, k } , thenin particular k (cid:54) = 0 so thatΓ kij = n (cid:88) (cid:96) =0 12 g k(cid:96) (cid:0) g i(cid:96),j + g j(cid:96),i − g ij,(cid:96) (cid:1) = n (cid:88) (cid:96) =1 12 g k(cid:96) (cid:0) g i(cid:96),j + g j(cid:96),i − g ij,(cid:96) (cid:1) , and hence for 0 / ∈ { i, j, k } we have Γ kij ( p , p ) = Γ kij ( p ) = 0. We conclude thatfor arbitrary ( i, j, k ) we have Γ kij ( p , p ) = 0. This completes the proof of LemmaA.1. (cid:3)
60 J.W. FISH AND H. HOFER
Corollary A.2 (properties of g and Γ) . In the coordinates as above, we have ∇ ∂ xi ∂ x j (cid:12)(cid:12) ( p ,p ) = 0 and ∇ ∂x i dx j (cid:12)(cid:12) ( p ,p ) = 0 . Proof.
The first equality follows from the fact that in our coordinate system, ∇ ∂ xi ∂ x j = Γ kij ∂ x k and the Γ kij vanish at ( p , p ) by Lemma A.1 above. To provethe second equality, we covariantly differentiate the equality δ ij = dx i ( ∂ x j ) to find0 = ( ∇ ∂ xk dx i )( ∂ x j ) + dx i ( ∇ ∂ xk ∂ x j ) . By our previous results, the second term vanishes when evaluated at ( p , p ). Sincethe above equality holds for each k ∈ { , . . . , n − } , and { ∂ x k } k ∈{ ,...,n − } forms abasis of T ( p ,p ) ( R × M ), we see that indeed ∇ ∂ xi dx j (cid:12)(cid:12) ( p ,p ) = 0 as claimed. (cid:3) Corollary A.3 ( ∂ a and da are parallel) . Let ( M, g ) be a Riemannian manifold, and consider the manifold R × M equippedwith the Riemannian metric ¯ g = da ⊗ da + g where a is the coordinate on R . Thenfor the Levi-Civita connection ∇ associated to ¯ g on R × M , we have ∇ da = 0 and ∇ ∂ a = 0 . Proof.
Let ( x , x , . . . , x n ) be coordinates as above and let V = v i ∂ x i be an arbi-trary smooth vector field. Then ∇ V ∂ a = v i ∇ ∂ ix ∂ x = v i Γ ki ∂ x k , where Γ kij = g k(cid:96) (cid:0) g i(cid:96),j + g j(cid:96),i − g ij,(cid:96) (cid:1) , and more importantly Γ ki = g k(cid:96) (cid:0) g i(cid:96), + g (cid:96),i − g i ,(cid:96) (cid:1) , = g k(cid:96) (cid:0) g (cid:96),i − g i ,(cid:96) (cid:1) , = g k(cid:96) (cid:0) g (cid:96),i (cid:1) , = 0 , where to obtain the second inequality we note that the metric ¯ g is R -invariant andhence ∂∂x g ij = 0; to obtain the third equality we have used that ¯ g i = δ i ; thefourth equality follows similarly. This establishes that ∇ ∂ a = 0. As noted above,we also have ∇ ∂ xk dx i = − Γ (cid:96)ki dx (cid:96) , which then establishes that ∇ da = 0. (cid:3) Definition A.4 (second fundamental form B ) . Let u : S → M denote an immersion into a Riemannian manifold ( M, g ) . Thenthe second fundamental form associated to u and g is denoted B u ∈ Γ( u ∗ ( T ∗ M ⊗ T ∗ M ⊗ T M )) , and is defined by B u ( X, Y ) := ( ∇ X Y ) ⊥ where X, Y ∈ Γ( u ∗ ( T M )) , ∇ denotes covariant differentiation with respect to theLevi-Civita connection associated to g , and Z (cid:55)→ Z ⊥ denotes orthogonal projectionto the normal bundle of u ( S ) . ERAL CURVES AND MINIMAL SETS 161
A.2.
Carefully Formulating the Co-Area Formula.
Given an oriented m -dimensional Riemannian manifold ( M, g ), there exists a canonical volume formgiven by ∗ (1), where ∗ is the Hodge ∗ -operator; we shall denote this form dµ mg .Recall that in local coordinates ( x , . . . , x m ) with ∂ x , . . . , ∂ x m a positive basis, wehave(137) dµ mg = (cid:113) det( g ij ) dx ∧ · · · ∧ dx m ;here g = g ij dx i ⊗ dx j . Although dµ mg is a volume form, we may regard it as ameasure via integration: dµ mg ( O ) := (cid:82) O dµ mg . We will abuse notation by letting dµ mg denote both the measure and volume form referring to each as needed.Before continuing on to establish some useful results, we make two last obser-vations. First, if S is oriented and u : S → M is an immersion, then ( S, u ∗ g )is an oriented Riemannian manifold. Second, if ( S, g ) is an oriented Riemannianmanifold of dimension k , and ω is a differentiable k -form on S , then we have thefollowing. (cid:90) S ω = (cid:90) S ω ( e , . . . , e k ) dµ ku ∗ g where ( e , . . . , e k ) forms a positive u ∗ g -orthonormal frame. This is straightforwardto verify. Proposition A.5 (The co-area formula) . Let ( S, g ) be a C oriented Riemannian manifold of dimension two; we allow that S need not be complete . Suppose that β : S → [ a, b ] ⊂ R is a C function withoutcritical points. Let f : S → [0 , ∞ ) be a measurable function with respect to dµ g .Then (138) (cid:90) S f (cid:107)∇ β (cid:107) g dµ g = (cid:90) ba (cid:16) (cid:90) β − ( t ) f dµ g (cid:17) dt where ∇ β is the gradient of β computed with respect to the metric g .Proof. We begin by defining two vector fields (cid:126)x = ( ∇ β ) / (cid:107)∇ β (cid:107) and (cid:126)y which isuniquely defined by the three conditions: (cid:107) (cid:126)y (cid:107) g = 1, (cid:104) (cid:126)x, (cid:126)y (cid:105) g = 0, and { (cid:126)x, (cid:126)y } is apositive basis. The flow of (cid:126)y preserves β since it is orthogonal to ∇ β , and dβ ( (cid:126)x ) = 1.Let ϕ t(cid:126)y and ϕ s(cid:126)x respectively denote the time t flow of (cid:126)y and the time s flow of (cid:126)x .For each ζ ∈ S , we then define the map Φ ζ : ˜ O ζ := ( − (cid:15) ζ , (cid:15) ζ ) × ( − (cid:15) ζ , (cid:15) ζ ) → S to be Φ ζ ( s, t ) = ϕ s(cid:126)x ( ϕ t(cid:126)y ( ζ )), with (cid:15) ζ > ζ is adiffeomorphism with its image, which we denote O ζ . Observe that β ◦ Φ − ζ ( x , x ) = x by construction.We now define functions ˜ f := f ◦ Φ ζ and ˜ β := β ◦ Φ ζ , and the metric ˜ g := Φ ∗ ζ g .In these local coordinates, we write ˜ g = ˜ g ij dx i ⊗ dx j , and then dµ g = (cid:113) det (˜ g ij ) dx ∧ dx = (cid:113) det (˜ g ij ) d ˜ β ∧ dx . Similarly, the volume form on the level sets of ˜ β are then given as dµ g = (cid:112) ˜ g dx That is, there may exist Cauchy sequences, with respect to g , which do not converge in S .
62 J.W. FISH AND H. HOFER
Making use of the fact that in our special case ˜ O ζ is a product space, we mayemploy Tonelli’s theorem to obtain (cid:90) ˜ O ζ ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g dµ g = (cid:90) ˜ O ζ ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g (cid:113) det (˜ g ij ) dx ∧ dx = (cid:90) ( − a ζ ,b ζ ) × ( − (cid:15) ζ ,δ ζ ) ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g (cid:113) det (˜ g ij ) dx dx = (cid:90) b ζ − a ζ (cid:16) (cid:90) δ ζ − (cid:15) ζ ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g (cid:113) det (˜ g ij ) dx (cid:17) dx = (cid:90) b ζ − a ζ (cid:16) (cid:90) ˜ β − ( t ) ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g (cid:112) det (˜ g ij ) √ ˜ g dµ g (cid:17) dt We now claim the following.(139) (cid:107)∇ ˜ β (cid:107) ˜ g = √ ˜ g (cid:112) det (˜ g ij )To see this is true we first note that it is sufficient to work pointwise. Next, we let v be a ˜ g -unit vector orthogonal to the level sets of ˜ β (i.e. orthogonal to the sets { x = const } ). It is elementary to show that v can be written as v = (cid:0) ˜ g ∂ x − ˜ g ∂ x (cid:1) / (cid:0)(cid:112) ˜ g (cid:113) det (˜ g ij ) (cid:1) . We then compute (cid:107)∇ ˜ β (cid:107) g = (cid:0) d ˜ β ( v ) (cid:1) = (cid:0) dx ( v ) (cid:1) = ˜ g ˜ g det (˜ g ij ) = ˜ g det (˜ g ij )and equation (139) is established. Consequently we have established (cid:90) O ζ f (cid:107)∇ β (cid:107) g dµ g = (cid:90) ˜ O ζ ˜ f (cid:107)∇ ˜ β (cid:107) ˜ g dµ g (140) = (cid:90) b ζ − a ζ (cid:16) (cid:90) ˜ β − ( t ) ˜ f dµ g (cid:17) dt = (cid:90) β ( ζ )+ b ζ β ( ζ ) − a ζ (cid:16) (cid:90) O ζ ∩ β − ( t ) f dµ g (cid:17) dt. We now prove the more general case by a partition of unity argument. First, foreach point ζ ∈ S , we let O ζ denote the open set containing ζ constructed above, andwe let Φ ζ : ˜ O ζ → O ζ denote the associated diffeomorphism. These diffeomorphismsshow S is locally compact. Since S is a manifold, it is second countable andHausdorff; together with being locally compact this guarantees S is paracompactand Hausdorff, and hence the open cover {O ζ } ζ ∈ S admits a subordinate partitionof unity { ρ α : S → [0 , } α ∈ I . That is, there is an index set I and an open cover {U α } α ∈ I of S , and there exist functions { ρ α } α ∈ I with the property that • supp( ρ α ) ⊂ U α ⊂ O ζ α , • for each ζ ∈ S we have { α ∈ I : ζ ∈ U α } < ∞ , • (cid:80) α ∈ I ρ α = 1. ERAL CURVES AND MINIMAL SETS 163
Now, making use of the partition of unity, equation (140), and the monotoneconvergence theorem to pass limits through integrals, we find the following. (cid:90) S f (cid:107)∇ β (cid:107) g dµ g = (cid:90) S (cid:88) α ∈ I ρ α f (cid:107)∇ β (cid:107) g dµ g = (cid:88) α ∈ I (cid:90) S ρ α f (cid:107)∇ β (cid:107) g dµ g = (cid:88) α ∈ I (cid:90) O ζα ρ α f (cid:107)∇ β (cid:107) g dµ g = (cid:88) α ∈ I (cid:90) β ( ζ α )+ b ζα β ( ζ α ) − a ζα (cid:16) (cid:90) O ζα ∩ β − ( t ) ρ α f dµ g (cid:17) dt = (cid:88) α ∈ I (cid:90) ba (cid:16) (cid:90) β − ( t ) ρ α f dµ g (cid:17) dt = (cid:90) ba (cid:16) (cid:90) β − ( t ) (cid:88) α ∈ I ρ α f dµ g (cid:17) dt = (cid:90) ba (cid:16) (cid:90) β − ( t ) f dµ g (cid:17) dt. This is the desired result, and this completes the proof of Proposition A.5. (cid:3)
A.3.
Typically Tame Perturbations.
The purpose of this section is to proveLemma A.8 below, which is the lemma which essentially proves the existence oftame perturbations via Lemma 4.26. In order to prove the main result here, it willbe useful to have the following definition established.
Definition A.6 (generally-Riemannian metric) . On a manifold S , which is smooth and may have boundary and corners, we call thepair ( Z , γ ) a generally-Riemannian metric provided Z ⊂ S \ ∂S is finite, γ is aRiemannian metric on S \ Z , and γ vanishes on Z . Remark A.7 (generally-Riemannian metrics yield distances) . Although a generally-Riemannian metric is not, strictly speaking, a Riemannianmetric, it nevertheless induces a distance function defined by the following. dist γ ( ζ , ζ ) := inf (cid:110) (cid:90) γ (cid:0) α (cid:48) ( t ) , α (cid:48) ( t ) (cid:1) dt : α ∈ C ([0 , , S ) and α ( i ) = ζ i (cid:111) . Lemma A.8 (sufficienty small perturbations) . Let (cid:15) (cid:48) , δ > be small positive constants. Let S be a compact real two-dimensionalmanifold possibly with boundary and possibly with corners. Suppose further that S is equipped with the following data.(1) h : S → R a smooth function satisfying { ζ ∈ ∂S : dh ( ζ ) = 0 } = ∅ ,(2) ( Z , γ ) a generally-Riemannian metric.Suppose δ satisfies δ < min (cid:16) dist γ ( Z , ∂S ) , min ζ , ζ ∈Z ζ (cid:54) = ζ dist γ ( ζ , ζ ) , dist γ (cid:0) { ζ ∈ S : dh ( ζ ) = 0 } , ∂S (cid:1)(cid:17) . Define the sets U δ = { ζ ∈ S : dist γ ( ζ, Z ) < δ }U δ = { ζ ∈ S : dist γ ( ζ, Z ) < δ }V δ = (cid:8) ζ ∈ S : dist γ (cid:0) ζ, { z ∈ S : dh ( z ) = 0 } (cid:1) < δ (cid:9) . Then there exists a function f ∈ C ∞ ( S ) satisfying the following conditions(f1) supp( f ) ⊂ V δ \ U δ with V δ ∩ ∂S = ∅ ,
64 J.W. FISH AND H. HOFER (f2) sup ζ ∈ Ω | f ( ζ ) | + sup ζ ∈ Ω (cid:107) df ( ζ ) (cid:107) γ + sup ζ ∈ Ω (cid:107)∇ df ( ζ ) (cid:107) γ < (cid:15) (cid:48) where Ω = supp( f ) , and ∇ denotes covariant differentiation associated to theLevi-Civita connection corresponding to γ ,(f3) on S \ U δ the function h + f is Morse; that is, the Hessian ∇ d ( h + f ) at criticalpoints of h + f is non-degenerate.Proof. We begin by regarding dh as a section of the cotangent bundle T ∗ S , so thezeros of dh are precisely the critical points of h , denoted byCrit h = { ζ ∈ S : dh ( ζ ) = 0 } . For each z ∈ Crit h \ Z , we may regard the linear map A z : T z S → T ∗ z SY (cid:55)→ ∇ Y dh (cid:12)(cid:12) z as the linearization of the principal part of the section dh ∈ Γ( T ∗ S → S ) atthe point z . Because T z S and T ∗ z S have the same dimension, we see that z is anon-degenerate critical point of h if and only if A z has trivial kernel. For each z ∈ Crit h \ Z we can define γ -geodesic coordinates ( x z , x z ) centered at z , with theadditional property that if A z has nontrivial kernel, then A z ( ∂ x z ) = 0. Next, foreach z ∈ Crit h \ Z we define the number n z ∈ { , , } by n z := dim ker( A z ), andwe define the neighborhood W z := (cid:8) ζ ∈ S : dist γ ( ζ, z ) < min (cid:0) δ , inj γ ( z ) (cid:1)(cid:9) , where inj γ ( z ) is the injectivity radius associated to γ at z ∈ S . Letting pr : S × R n z → S denote the canonical projection to the first factor, we define thesection σ z ∈ Γ(pr ∗ T ∗ S → S × R n z ) by σ z ( ζ, s ) = (cid:40) dh ( ζ ) if ζ / ∈ W z or n z = 0 dh ( ζ ) + (cid:80) n z i =1 s i d ( x iz β z )( ζ ) otherwisewhere ( x z , x z ) are the coordinates established above, and β z is a smooth cut-offfunction with β z ( ζ ) = 1 in a neighborhood of ζ = z and(141) supp( β z ) ⊂ (cid:8) ζ ∈ S : dist γ ( ζ, z ) < min( δ , inj γ ( z )) (cid:9) . By construction the section σ z is transverse to the zero section at the point ( z, z ∈ Crit h \Z , and hence the linearization of the principal part of σ z is surjectiveat ( z, O z ⊂ S containing the point z with the additional property that for each ( w, ∈ O z × R n z the linearization of theprincipal part of σ z at ( w,
0) is surjective. We now repeat this construction for each z ∈ Crit h \ Z . Note that Crit h \ U δ is compact, and the collection {O z } z ∈ Crit h \Z is an open cover of Crit h \ U δ , and hence may be reduced to a finite sub-cover of By “linearization of the principal part” we mean the following. Given a vector bundle
E → B with a connection T E = V E ⊕ H E , which for our purposes will always be the Levi-Civitaconnection, together with a continuously differentiable section σ : B → E , then the linearizationof the principal part of σ is defined to be pr V ◦ T σ where pr V : T E → V E is the projection tovertical sub-bundle associated to the connection. ERAL CURVES AND MINIMAL SETS 165
Crit h \ U δ , which we denote by {O z , . . . , O z m } . Let { w , . . . , w (cid:96) } ⊂ { z , . . . , z m } be the subset for which A w k fails to be surjective, and define the section ˜ σ by˜ σ ∈ Γ (cid:0) pr ∗ T ∗ S → S × R n w × · · · × R n w(cid:96) (cid:1) ˜ σ ( ζ, s ) = dh ( ζ ) + (cid:96) (cid:88) j =1 n wj (cid:88) i =1 s j,i d ( x iw j β w j )( ζ )For ease of notation, let us re-index and rewrite the above as˜ σ ∈ Γ (cid:0) pr ∗ T ∗ S → S × R ˜ m (cid:1) ˜ σ ( ζ, s ) = dh ( ζ ) + ˜ m (cid:88) i =1 s i d ˜ f i ( ζ ) . By construction, the linearization of the principal part of the section ˜ σ is surjectiveover the set O × { } where O = ∪ mi =1 O z i . Consequently, there exists an open set (cid:101) O = O × {| s | < ˆ (cid:15) } ⊂ S × R ˜ m which has the following two important properties.(T1) the linearization of the principal part of ˜ σ is surjective at every point in (cid:101) O (T2) if ( ζ, s ) ∈ S × {| s | < ˆ (cid:15) } solves ˜ σ ( ζ, s ) = 0, then ζ ∈ O ∪ U δ The first property follows essentially because surjectivity is an open condition; thesecond property follows because dh is non-vanishing on the compact set S \ ( O∪U δ ),so that (cid:107) dh (cid:107) attains a non-zero minimum on this set, and hence for all s sufficientlyclose to 0 we have sup ζ ∈ S (cid:107) ˜ m (cid:88) i =1 s i d ˜ f i ( ζ ) (cid:107) γ < inf ζ ∈ S \ ( O∪U δ ) (cid:107) dh ( ζ ) (cid:107) . As a consequence of property (T1) and the implicit function theorem, the set B := (cid:101) O ∩ ˜ σ − (0) ⊂ S × R ˜ m is a smooth manifold of dimension ˜ m . Fix ( ζ, s ) ∈ B , andnote that the associated tangent fiber of B is given by T ( ζ, s ) B = { ( v, ˆ s , . . . , ˆ s ˜ m ) ∈ T ζ S × R ˜ m : 0 = ∇ v dh + ˜ m (cid:88) i =1 s i ∇ v d ˜ f i + ˆ s i d ˜ f i } . We denote the following vector spaces X = T ζ S , Y = T ∗ ζ S , and Z = R ˜ m , and wedefine the following linear maps. D : X → Y by D ( v ) = ∇ v dh + ˜ m (cid:88) i =1 s i ∇ v d ˜ f i L : Z → Y by L (ˆ s , . . . , ˆ s ˜ m ) = ˜ m (cid:88) i =1 ˆ s i d ˜ f i Consequently, we may express T ( ζ, s ) B = ker ( D ⊕ L ) . Finally, we define the projectionΠ : ker ( D ⊕ L ) → Z by Π( v, ˆ s , . . . , ˆ s i ) = (ˆ s , . . . , ˆ s i ) . At this point we note that D ⊕ L is the linearization of the principal part of ˜ σ atthe point ( ζ, s ), which by construction is surjective, and hence D ⊕ L is onto. ByLemma A.9 below, it follows that D is surjective if and only if Π is surjective.
66 J.W. FISH AND H. HOFER
At this point, our aim is to show that there exist many choices of s ∈ R ˜ m withthe property that whenever ˜ σ ( ζ, s ) = 0, we also have that Π is surjective. Tothat end, consider pr : B ⊂ S × R ˜ m → R ˜ m the canonical projection to the secondfactor, and observe that this map is smooth, and T pr = Π. By Sard’s theorem, theregular values of pr have full measure, and hence there exists a sequence { s k } k ∈ N in R ˜ m satisfying s k → s k is a regular value of pr . For each such s k and every ( ζ, s k ) ∈ B we then have that D is surjective. That is to say, foreach fixed such s k = ( s k , . . . , s ˜ mk ), and each ζ ∈ S which is a zero of the section d ( h + f k ) := d ( h + (cid:80) i s ik ˜ f i ) ∈ Γ( T ∗ S → S ), the linearization of the principalpart of this section is surjective. In other words, for each such s k , the section d ( h + (cid:80) i s ik ˜ f i ) is transverse to the zero-section for all ζ ∈ O ; by property (T2)the only zeros of d ( h + f k ) lie in O ∪ U δ , and hence all critical point of h + f k in S \ U δ are non-degenerate. This establishes property (f3) for any f = f k . Nextwe note that f k → C ∞ so that property (f2) holds for any sufficiently large k .Finally property (f1) follows from the definition of the f k – specifically the supportof the cut-off functions β z established in equation (141). This completes the proofof Lemma A.8 (cid:3) Lemma A.9 (Lemma A.3.6, [30]) . Assume D : X → Y is a Fredholm operator and L : Z → Y is a bounded linearoperator such that D ⊕ L : X ⊕ Z → Y is onto. Then D ⊕ L has a right inverse.Moreover, the projection Π : ker( D ⊕ L ) → Z is a Fredholm operator with ker Π ∼ =ker D and coker Π ∼ = coker D , and hence index Π = index D . References [1] Casim Abbas and Helmut Hofer,
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Joel W. Fish, Department of Mathematics, University of Massachusetts Boston
E-mail address : [email protected] Helmut Hofer, School of Mathematics Institute for Advanced Study
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