Almost Existence From the Feral Perspective and Some Questions
aa r X i v : . [ m a t h . S G ] J un ALMOST EXISTENCE FROM THE FERAL PERSPECTIVE ANDSOME QUESTIONS
JOEL W. FISH AND HELMUT H. W. HOFER
Contents
1. Introduction and Historical Background 11.1. The Weinstein Conjecture 21.2. Almost Existence 31.3. Statement of the Main Result 51.4. Speculation on the C ∞ -Closing Property 82. Background and Geometric Framework 93. Proof of the Main Result 17Appendix A. Miscellaneous Support 34A.1. Supporting Proofs 34A.2. Extra Definitions 40References 401. Introduction and Historical Background
This paper is concerned with the “ almost existence ” phenomenon for periodicorbits of Hamiltonian dynamical systems. We shall describe some of the backgroundof this phenomenon, and we relate it to the new feral curve theory, [8], which wasrecently initiated by the authors. The new approach to the “ almost existence ”phenomenon suggests a larger context which also features questions around the C ∞ -closing lemma in the following sense. Consider a compact symplectic manifold( W, Ω) equipped with a smooth Hamiltonian H : W → R . Given a regular energysurface H − ( E ), one can ask the question: Is it the case that after a small smoothperturbation of H the new H ′− ( E ) has the property that the set of periodic orbitsare dense? This is a classical question, and the C -closing lemma shows that theabove assertion is true for a C -small perturbation of the Hamiltonian, i.e. for a C -close Hamiltonian vector field. Whereas the C -closing lemma holds in general, it isknown that the C ∞ -closing lemma does not; see [11]. We believe that the analysisbelow contributes to a growing body of evidence which suggests that the validityof the Hamiltonian C ∞ -closing lemma is intimately connected to the existence ofa sufficiently rich Gromov-Witten theory of the ambient space. We touch on thisspeculation in Section 1.4. Date : July 1, 2020.
Key words and phrases. feral curves, almost existence, adiabatic.The first author’s research in development of this manuscript was supported in part by NSF-DMS Research Grant Award 1610452.
The main goal of this paper is to set up a body of results to utilize the FeralCurve theory to study questions around almost existence and the closing lemma.Our results show that given a compact pile of Hamiltonian energy surfaces, a suf-ficient supply of pseudoholomorphic curves associated to this pile implies “almostexistence”; see Theorem 3. This is always attainable provided the pile of energysurfaces can be viewed as lying in a compact symplectic manifold with a sufficientsupply of pseudoholomorphic curves. As an exercise the reader might enjoy usingour more local results to reprove the almost existence result for regular compactenergy surfaces in R n by using the large supply of pseudoholomorphic curves ob-tained as a deformation of holomorphic curves in CP n . Building on previous workin [8], we also show that under suitable topological constraints, the almost existenceresult can be improved to the following: Every energy level contains a non-trivialclosed invariant subset, and for almost all of these energy levels this set is a peri-odic orbit; see Theorem 3. For the convenience of the reader we recall the necessarybackground.1.1. The Weinstein Conjecture.
As a starting point for our discussion, we con-sider two seminal papers by P. Rabinowitz, namely [22] and [23], which are con-cerned with the existence of a periodic orbit on a given compact regular energysurface M = H − ( E )for an autonomous Hamiltonian system defined on the standard phase space R n .Rabinowitz showed the existence of periodic orbits whenever suitable geometricconditions are met. We refer the reader to the introduction of [1] for the interestingbroader historical perspective of Rabinowitz’ results.A. Weinstein analyzed these results, particularly [23], and proposed the far-reaching Weinstein Conjecture in [29], which we shall describe momentarily. Firstthough, we provide some definitions. Consider an odd-dimensional smooth manifold M n +1 , with a one-form λ for which λ ∧ ( dλ ) n is a volume form. In this case, wecall λ a contact form for M , which then uniquely determines a vector field X bythe following equations: λ ( X ) ≡ i X dλ ≡ . In this case, we call X the Reeb vector field associated to the contact form λ . In amodified form, the Weinstein Conjecture states the following. Weinstein Conjecture (1978):
Let M be a smooth closed odd-dimensional manifold equipped with a contact formand an associated Reeb vector field X . Then the dynamical system given by ˙ x = X ( x ) , has a nontrivial periodic orbit. The first breakthrough concerning the Weinstein Conjecture was Viterbo’s cel-ebrated result in [28], which showed that a regular compact contact-type energysurface of a Hamiltonian system in R n carries a nontrivial periodic orbit. In 1993Hofer, in [13], showed that for a contact form λ on a closed three-manifold M theWeinstein conjecture is true provided at least one of the following holds. Either LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 3 M = S , or π ( M ) = 0, or λ is overtwisted. In 2007 Taubes proved in [27] that theWeinstein conjecture in dimension three is true for all ( M, λ ).A natural question then becomes whether or not a generalization of this resultholds for more general compact energy surfaces in R n . However, if n ≥ n = 2, see V. Ginzburg and B. G¨urel, [10], and thus itbecomes interesting to study, in some sense, how often and how the generalizationfails. This is the content of Section 1.2.1.2. Almost Existence.
By analyzing Viterbo’s paper, the second author and E.Zehnder established in [16] that many compact Hamiltonian energy levels containperiodic orbits. Indeed, after some refinement by Rabinowitz in [24] and Struwein [25], this phenomenon became referred to as “ almost existence ”. Working ina context in which the almost existence phenomenon holds, the actual existencequestion for periodic orbits of Hamiltonian systems quite often can be phrased as“a priori estimates imply existence”, see [4]. Finally the phenomenon was explainedin [18] in terms of differentiability properties of the so-called Hofer-Zehnder capacity,see also [17].For our explicit purpose of connecting this topic to feral curve theory, we ap-proach the subject in a particular way. In a first definition we give an abstractionof a regular, smooth and compact Hamiltonian energy surface in a symplectic man-ifold, which we call a “ framed Hamiltonian manifold ”. By forgetting some of theinformation carried by a framed Hamiltonian manifold we obtain what is called an“ odd-symplectic manifold ”. Definition 1.1 (framed Hamiltonian manifold and odd-symplectic) . A framed Hamiltonian manifold ( M, λ, ω ) consists of a smooth closed odd-dimen-sional manifold M = M n +1 , a two-form ω and a one-form λ such that λ ∧ ω n isa volume form. When such a λ exists but we only specify ( M, ω ) , we call the pairan odd-symplectic manifold. A framed Hamiltonian manifold (
M, λ, ω ) defines a dynamical system. Namelythere exists a non-singular vector field X on M uniquely characterized by theequations i X λ ≡ i X ω ≡ . As in the more special contact case we shall refer to X as the Reeb vector field (associated to (
M, λ, ω )).Assume that we are given a symplectic manifold ( W, Ω) without boundary andconsider a compact, smooth, regular and co-oriented hypersurface M in W . De-noting by ι : M → W the inclusion, we abbreviate ω = ι ∗ Ω and obtain theodd-symplectic manifold (
M, ω ). If H : W → R is a smooth Hamiltonian and H − ( E ) = M for some number E ∈ R and dH ( m ) = 0 for m ∈ M we can take aone-form λ on M such that λ ( X H ( m )) = 1 for m ∈ M . Then λ ∧ ω n is a volumeform on M . Hence we obtain a framed Hamiltonian manifold ( M, λ, ω ). One easilyverifies that the Reeb vector field X satisfies X H ( m ) = X ( m ) for all m ∈ M, where X H is the Hamiltonian vector field associated to H and defined by i X H Ω = − dH . J.W. FISH AND H. HOFER
In order to study the almost existence phenomenon we also need to considerneighborhoods of a smooth, regular and compact energy surface. Given a co-orientable, compact, smooth and regular hypersurface M contained in W , where( W, e Ω) is a symplectic manifold, we obtain an odd-symplectic manifold (
M, ω ) aspreviously described. Namely we take the inclusion i : M → W and define ω := i ∗ e Ω.We fix a one-form λ such that λ ∧ ω n is a volume form on M . We can define on R × M with coordinates ( t, m ) a 2-form Ω byΩ = d ( tλ ) + ω. It is a trivial exercise that there exists an open neighborhood U of M ≡ { } × M such that Ω | U is a symplectic form. Moreover, if we take U small enough wefind an embedding ψ : U → W onto an open neighborhood of M ⊂ W such that ψ ∗ e Ω = Ω | U and in addition ψ (0 , m ) = m for all m ∈ M . Note that this also impliesthat given λ and λ so that λ i ∧ ω n are volume forms we find for the correspondingΩ i defined by Ω i = d ( tλ i ) + ω, open neighborhoods U and U of M ⊂ R × M such that Ω i | U i are symplectic andthere exists a symplectic diffeomorphism ψ : U → U , which is the identity on M = { } × M . Depending on the case, whether or not λ ∧ ω n or λ ∧ ω n definethe same orientation on M , we must have that ψ maps ( s, m ) for some s > s ′ , m ′ ) with ± s ′ > U of M we may assume that U = R × M equipped with asymplectic form Ω such that ω = ι ∗ Ω, where ι ( m ) = (0 , m ) for m ∈ M . Definition 1.2 (compatible Ω) . Given an odd-symplectic ( M, ω ) we call a symplectic form Ω on R × M compatibleprovided ι ∗ Ω = ω , where ι : M → R × M is defined by ι ( m ) = (0 , m ) for m ∈ M . The notion of “almost existence” will be associated to the behavior of smallneighborhoods of M = { } × M in R × M for a symplectic form which restricts to ω . Definition 1.3 (almost existence property - odd symplectic) . Consider a smooth, compact odd-symplectic manifold ( M, ω ) , denote by Ω a com-patible symplectic form on R × M and identify M ≡ { } × M . We say that ( M, ω ) has the almost existence property provided there exists an open neighborhood U of M with the following property. Given any proper, smooth and surjective Hamiltonian H : V → ( − ε, ε ) for some ε > , where V is an open neighborhood of M containedin U such that H − (0) = ι ( M ) and dH ( s, m ) = 0 for m ∈ M and s ∈ ( − ε, ε ) definethe set Σ H by Σ H = { E ∈ ( − ε, ε ) | ˙ x = X H ( x ) has a periodic orbit with H ( x ) = E } . We say that ( M, ω ) has the almost existence property provided for suitable U itholds for all H as described above that measure (Σ H ) = 2 · ε . Finally we can introduce a special class of symplectic manifolds.
Definition 1.4 (almost existence property – symplectic manifold) . A symplectic manifold ( W, Ω) without boundary has the almost existence prop-erty provided for every regular, compact, and co-oriented hypersurface M the pair ( M, ι ∗ Ω) has the almost existence property, where ι : M → W is the inclusion. LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 5
We know that the standard symplectic vector space R n has the almost existenceproperty, see [17]. We also know that T can be be equipped with a symplecticform so that ( T , Ω) does not have the almost existence property, see also [17]. Inother words, the almost existence property is nontrivial. We state the followingtheorem for the convenience of the reader. It is based on some known facts whichwe identify as a local property.
Theorem 1 (local almost existence property) . Every symplectic manifold ( W, Ω) without boundary has the following property.Given a point w ∈ W there exists an open neighborhood U ( w ) so that for everyclosed regular hypersurface M ⊂ U the pair ( M, i ∗ Ω) has the almost existence prop-erty. In other words, every symplectic manifold without boundary has the ”local al-most existence property”. It then becomes an interesting question what kind ofmore global compact, regular hypersurfaces in ( W, Ω) have the almost existenceproperty. We formulate a more precise question next.
Question 1.5.
Assume that ( W, Ω) is a symplectic manifold without boundary.Suppose that M ⊂ W is a smooth, compact, regular hypersurface without boundaryso that the inclusion ι : M → W is isotopic to a small hypersurface, i.e. containedin some U ( w ) , see Theorem 1. Does then M have the almost existence property?If this is not always true, then for which class of symplectic manifolds (other than R n ) is it true? We note that the literature suggests that compact symplectic manifolds with asufficiently rich Gromov-Witten theory have the almost existence property. Thefirst paper, predating Gromov-Witten theory, where such an idea is used is [15]. Itshows that having a suitable moduli space of pseudoholomorphic curves associatedto a symplectic manifold implies that the Weinstein conjecture holds for suitableenergy surfaces. The method in [15] was then used in [20] and combined withGromov-Witten theory for a more convenient packaging of properties of modulispaces. This Gromov-Witten style approach has also been used to prove almostexistence results in certain cases; see for example [21], Theorem 1.10.1.3.
Statement of the Main Result.
Here we state the main result of the article.The terms used below are standard, however they are provided explicitly in Section2 below. For example, the notion of an Ω-tame almost complex structure is providedin Definition 2.1; the notion of a pseudoholomorphic map is provided in Definition2.2; and the notion of the genus of such a map is provided in Definition 2.3.
Theorem 2 (Main Result) . Let ( W, Ω) be a symplectic manifold without boundary, and let H : W → R be asmooth proper Hamiltonian. Fix E − , E + ∈ H ( W ) ⊂ R with E − < E + , as wellas positive constants, C g > , and C Ω > . Suppose that for each Ω -tame almostcomplex structure J on W there exists a proper pseudoholomorphic map u : ( S, j ) → { p ∈ W : E − < H ( p ) < E + } without boundary, which also satisfies the following conditions: By proper here, we mean that for each compact set
K ⊂ R , the set H − ( K ) is compact. J.W. FISH AND H. HOFER (1) (genus and area bounds)
The following inequalities hold:
Genus( S ) ≤ C g and Z S u ∗ Ω ≤ C Ω . (2) (energy surjectivity) The map H ◦ u : S → ( E − , E + ) is surjective.Then there is a periodic Hamiltonian orbit on almost every energy level in range ( E − , E + ) . That is, if we let I ⊂ ( E − , E + ) denote the energy levels of H whichcontain a Hamiltonian periodic orbit, then I has full measure: µ ( I ) = µ (cid:0) ( E − , E + ) (cid:1) = E + − E − . At this point there are a few points worth making. The first is that the pseu-doholomorphic maps in question here need not be compact – in fact, a carefulinspection of the requirements reveals that they cannot be compact.A second point is that the pseudoholomorphic maps allowed by the above hy-potheses may have domains (
S, j ) which are diffeomorphic to an open annulus, butsometimes the domains will be much worse. For example, our assumptions allowfor the possibility that a domain of a pseudoholomorphic map may be an infinitelypunctured open disk with infinitely many closed disks removed. Or worse: the opendisk with the Cantor set removed. Or any closed Riemann surface with any closedset removed. To be clear, such assumptions are highly unusual in the standard the-ory of pseudoholomorphic curves, however they are standard in feral curve theory.This is because feral curves in general are much wilder. We illustrate this with aplausible example.Suppose ( W, Ω) is a closed symplectic manifold, with an Ω-compatible almostcomplex structure J , a smooth Hamiltonian H : W → R for which 0 is a regularvalue, and a pseudoholomorphic map ¯ u : S → W where S is a closed Riemannsurface such as a sphere or torus, etc. For ǫ >
0, what structure does the followingset have? S := (cid:8) z ∈ S : − ǫ < H (cid:0) ¯ u ( z ) (cid:1) < ǫ (cid:9) (1)As it need not be the case that ± ǫ are regular values of H ◦ u , there is no reasonthat S should necessarily admit a smooth compactification to a compact Riemannsurface with smooth boundary. Indeed, all one can really say is that it has thestructure of a closed Riemann surface with some closed set removed, which is exactlythe sort of domain surface that Theorem 2 allows. Conceptually, it may be easierto think of the maps we allow as arising from restrictions like u := ¯ u (cid:12)(cid:12) S , with S asin equation (1), however it is worth noting that the existence of such an extension(or lack thereof) plays no role in our proof.With such unusual freedom allowed for the domains of our pseudoholomorphicmaps, the attentive reader may be concerned about the precise notion of genus.This is made rigorous in Definition 2.3 below, although it amounts to exhausting S by compact two-dimensional manifolds with boundary and taking the limit ofassociated the genera.A third point is that the hypothesis in Theorem 2 regarding the existence of asuitable pseudoholomorphic map for each Ω-tame almost complex structure seemsvery restrictive, however in practice this is not the case. For example, Gromov-Witten invariants are invariants of a closed symplectic manifold obtained by alge-braically counting pseudoholomorphic maps of specified genus, homology class, andincidence conditions. In particular, Gromov-Witten invariants are independent of
LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 7 the choice of almost complex structure (provided that it is Ω-tame), and thus asufficiently rich Gromov-Witten theory for a closed symplectic manifold ( W, Ω) issufficient to guarantee the hypotheses of Theorem 2 are satisfied in many cases.We will contrast the feral curve techniques used below with the methods usedin [20] and [21] momentarily, however at present we point out that the the proof ofTheorem 2 will show that its conclusion holds under weaker assumptions regardingthe pseudoholomorphic curves. Specifically, one only needs the existence of pseudo-holomorphic curves for a particular sequence of adiabatically degenerating almostcomplex structures. The notion of this adiabatic degeneration is rather technicaland is made precise in Definition 2.8, however the idea is to degenerate the almostcomplex structure so as to geometrically “stretch the neck” along a continuum ofenergy levels. With such an concept internalized, we direct the reader’s attentionto Theorem 4 in Section 3 below, which is a localized version of Theorem 2 withthe assumptions stripped to the absolute essentials.With the exception of Struwe’s results in [26], which predate modern symplecticmethods, all proofs of almost existence follow a similar pattern: prove the Hofer-Zehnder capacity of a domain containing the energy surface is finite, and the desiredresult follows from [18]. For example, in [21] Lu takes cues from [20] to use theGromov-Witten invariants to define a pseudo- capacity which is finite and boundsthe Hofer-Zehnder capacity; the almost existence result is then immediate. Incontrast, the proof of Theorem 2 makes no use of capacities at all, and only makesuse of pseudhoholomorphic curves – specifically feral curve theory. The idea isto stretch the neck along a continuum of energy levels, and analyze some basicproperties in the limit. For those familiar with methods from Symplectic FieldTheory, it is worth pointing out that the key obstacle to overcome is that there areno Hofer energy bounds in this case, and hence infinite energy pseudoholomorphiccurves (i.e feral curves) can appear in the limit “building.” The picture that emergesfrom this analysis is rather interesting. It seems that as one takes this adiabaticdegeneration, curves which cross the region of degeneration are inexorably drawnto collapse onto families of periodic orbits. Or more precisely, such a collapse tofamilies of orbits occurs almost everywhere, and on the complementary measurezero set the curves are allowed to jump between such families, or even jump to amore general closed invariant subset. Further analysis of such limiting curves seemswell poised to illuminate additional dynamical features.We close this introductory section with an application which appears to be in-accessible to methods relying on the finiteness of the Hofer-Zehnder capacity. Westate this as Theorem 3 below, but first provide a definition.
Definition 1.6 (positive contact type) . Let ( W, Ω) be a compact symplectic manifold with boundary. Let M + be a unionof connected components of ∂W . We say M + ⊂ ∂W is of positive contact type provided there exists an outward pointing nowhere vanishing vector field Y defined ina neighborhood of M + in W for which L Y Ω = Ω ; here L denotes the Lie derivative.In this case λ := i Y Ω (cid:12)(cid:12) M + is a contact form on M + . Theorem 3 (intertwining existence and almost existence) . Let ( W, Ω) be a four-dimensional compact connected exact symplectic manifold withboundary ∂W = M + ∪ M − . Suppose M + is positive contact type in the sense ofDefinition 1.6, and suppose that one of the following three conditions holds:(1) M + has a connected component diffeomorphic to S , J.W. FISH AND H. HOFER (2) there exists an embedded S ⊂ M + which is homotopically nontrivial in W ,(3) ( M + , λ ) has a connected component which is overtwisted.Then for each Hamiltonian H ∈ C ∞ ( W, R ) for which H − ( ±
1) = M ± , the fol-lowing is true. For each s ∈ [ − , the energy level H − ( s ) contains a closednon-empty set other than the energy level H − ( s ) itself which is invariant underthe Hamiltonian flow of X H ; moreover for almost every s ∈ [ − , this closedinvariant subset is a periodic orbit. Note that the image of H is not required to lie in [ − , Speculation on the C ∞ -Closing Property. Before moving on to the proofsof Theorem 2 and Theorem 3, we wish to draw some connections to the C ∞ -closinglemma. We also aim to pose some speculative questions which we believe the feralcurve techniques below seem well poised to eventually answer.We begin with an odd-symplectic manifold ( M, ω ), and take a compatible sym-plectic form Ω on R × M . We consider the Fr´echet space C ∞ ( M, R ) and observethat every element f defines a hypersurface in R × M by setting M f = { ( f ( m ) , m ) ∈ R × M : m ∈ M } . Considering the hypersurface M f ⊂ ( R × M, Ω) we obtain the distinguished linebundle L f → M f , given by L f := ker( ω f ) ⊂ T M f , where ω f is the pull-back of Ω by the inclusion M f → R × M . Since L f ⊂ T M f isa dimension one sub-bundle, it is an integrable distribution and we are interestedin the closed leaves. We denote by C f the union of all points in M f which lie on aclosed leaf. We say that the periodic orbits are dense on M f provided cl( C f ) = M f . Definition 1.7 ( C ∞ -closing property) . We say that the odd-symplectic manifold ( M, ω ) has the C ∞ -closing property pro-vided that there exists a compatible Ω on R × M so that for a Baire subset Σ of C ∞ ( M, R ) the following holds: cl( C f ) = M f for all f ∈ Σ . Again one can use the closing property to define a particular class of symplecticmanifolds.
Definition 1.8 ( C ∞ -closing property – symplectic manifolds) . We say a symplectic manifold ( W, Ω) has the C ∞ -closing property provided forevery regular compact co-oriented hypersurface M in W the induced ( M, ω ) has the C ∞ -closing property. One can play around with the above definition by allowing only hypersurfacesisotopic to small ones or those carrying a suitable topology. Alternatively, onemight choose to only allow contact-type hypersurfaces. We leave it to the readerto explore these ideas and we only mention the following conjecture.
Conjecture 1 (local C ∞ -closing property) . The standard symplectic vector space ( R n , Ω standard ) , n ≥ , has the C ∞ -closingproperty. In particular all symplectic manifolds without boundary have the local C ∞ -closing property. LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 9
The conjecture is open for all n ≥ n ≥
3. Inthe case of n = 2 one knows a partial result, namely that compact, regular hyper-surfaces in R of contact type have the C ∞ -closing property. However, nothing isknown for general compact regular hypersurfaces in R . Indeed, by a result of Irie,[19], every ( M, λ, dλ ), where M is a closed three-manifold equipped with a contactform λ , has the C ∞ -closing property. In particular every compact regular hypersur-face of contact type in R has the C ∞ -closing property. In the background of Irie’sresult and a follow-up result, [2], is the important volume formula by Cristofaro-Gardiner, Hutchings, and Ramos, [5]. At present the proof of [19] based on [5] onlyworks in the three-dimensional case. Due to the use of Seiberg-Witten-Floer The-ory it will need some new ideas to attack the higher-dimensional cases – perhapsferal curves.The germ of an idea goes as follows. The key upshot of feral curves, is that onecan stretch the neck along any hypersurface. The downside is that one may findlimit sets which are much more complicated than a finite set of periodic orbits.However, Theorem 2 strongly suggests that generically (in the right sense) one canstretch the neck while following a single curve for each almost complex structure,and pass to the limit to find a periodic orbit. That is, at least one periodic orbitis (generically) found by tracking just one curve. What if there are many curvesto track? High dimensional families of curves, for example. Here the proposedrichness of the Gromov-Witten invariants comes into play. For example, consider R n ≃ C n = C P n \ C P n − , and suppose we consider stretching the neck alongsome generic hypersurface M ⊂ R n . By considering curves of high degree, oneobtains high dimensional families of curves which stretch and break along periodicorbits. This raises a question: Which orbits can be found by stretching the neckand tracking curves of any fixed (but arbitrarily large) degree? All orbits, or justa subset? A dense subset? If it turns out that neck stretching can find (nearly)every orbit then feral curves seem well poised to recover the C ∞ -closing lemmafor arbitrary regular, compact hypersurfaces in R n . Currently the known resultsabout the contact-type case in R depend on Seiberg-Witten theory. The feralcurve theory should be important in removing the contact-type hypothesis. Inorder to prove the results in higher dimensions one way to succeed seems to be thedevelopment of suitable techniques to use higher-dimensional moduli spaces.2. Background and Geometric Framework
Here we will recall some standard background material, and then we providea geometric framework very specific to the problem of study. We begin with thenotions of an almost complex manifold and pseudoholomorphic maps.
Definition 2.1 (almost complex structures; compatible and tame) . Let W be a smooth manifold, not necessarily closed, possibly with boundary, andlet J ∈ Γ(End(TW)) be a smooth section for which J ◦ J = − . We call J analmost complex structure for W , and the pair ( W, J ) an almost complex manifold.In the case that Ω is a symplectic form on W , we say J is Ω -compatible providedthat g ( v, w ) := Ω( v, Jw ) is a Riemannian metric. Under the weaker assumptionthat Ω( v, Jv ) ≥ for all v ∈ T W , and with equality if and only if v = 0 , we sayinstead that J is only Ω -tame. Definition 2.2 (pseudoholomorphic map) . Let ( S, j ) and ( W, J ) be smooth almost complex manifolds with dim( S ) = 2 , each possibly with boundary. A C ∞ -smooth map u : S → W is said to be pseudoholo-morphic provided J · T u = T u · j . That is, the tangent map of u intertwines thealmost complex structures on domain and target. Unless otherwise specified, weallow S to be disconnected. We say such a map is proper provided the pre-imageof any compact set is compact. It is worth noting that constant maps are always pseudoholomorphic. We alsonote that we will allow the domains of our pseudoholomorphic maps to be discon-nected, which in conjunction with the fact that constant maps are always pseudo-holomorphic allows for the possibility that any given map may have many constantcomponents – perhaps infinitely many. At present we allow this, while noting thatsecond countability of the domain Riemann surfaces forces any given map to haveat most countably many constant components.Because the domains of our pseudoholomorphic maps are Riemann surfaceswhich need not be compact, we make the notion of genus precise with the fol-lowing definition.
Definition 2.3 (Genus) . Let S be a two-dimensional oriented manifold, possiblywith boundary, with at most countably many connected components, and with theproperty that each connected component 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 e S = (cid:0) S ⊔ ( ⊔ nk =1 D ) (cid:1) / ∼ to be the closed surface capped off by n disks, anddefine Genus( S ) := Genus( e 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 ) . We now turn our attention to geometric structures more specific to the proof ofTheorem 2. The first key idea is what we call a framed Hamiltonian energy pile (see Definition 2.4) which is essentially the neighborhood of a compact energy levelwith enough structure to regard it as something like a family of framed Hamiltonianmanifolds. This relationship is made precise with Lemma 2.6. Also important isLemma 2.5, which is a means to identify the neighborhood of a compact energylevel in a general symplectic manifold, with the structure of a framed Hamiltonianenergy pile, thereby localizing the almost existence problem.
Definition 2.4 (framed Hamiltonian energy pile) . Let I ǫ be the interval ( − ǫ, ǫ ) equipped with the coordinate s , and let M be a closedodd-dimensional manifold. Assume I ǫ × M is equipped with the symplectic form Ω . Let H : I ǫ × M → R be the smooth Hamiltonian H ( s, p ) = s , and let X H bethe associated Hamiltonian vector field determined by i X H Ω = − dH and let ˆ λ be aone-form on I ǫ × M which satisfies the following three conditions:(H1) ˆ λ ( ∂ s ) = 0 ,(H2) L ∂ s ˆ λ = 0 where L is the Lie derivative,(H3) ˆ λ ( X H ) > . LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 11
We call the triple ( I ǫ × M, Ω , ˆ λ ) a framed Hamiltonian energy pile. If we consider local coordinates x , ..., x N on M , N = 2 n + 1, and the coordinate s on I ε then ˆ λ can be written at the point ( s, x ) asˆ λ ( s, x ) = N X i =1 a i ( x ) dx i . (2)due to the imposed conditions (H1) and (H2). We shall show in Lemma 2.5 howframed Hamiltonian energy piles arise near a compact regular energy surface. Atfirst, however, we begin by deriving a few geometric structures which arise as animmediate consequence of having a framed Hamiltonian energy pile. We illuminatethem at present. Define the two-plane distribution ˆ ρ on I ε × M byˆ ρ = Span( ∂ s , X H ) , (3)and define the codimension-two plane distribution ξ on I ε × M by ξ = ker ( ds ∧ ˆ λ ) = (ker ds ) ∩ (ker ˆ λ ) . (4)We observe that the vector bundle ξ → I ε × M is R -invariant in the following sense.Given s , s ∈ I ε , the map ( h + s , m ) → ( h + s , m ), which is defined for small | h | ,pushes forward (the obvious restrictions of) ξ to ξ . Moreover, we have the splitting T ( I ǫ × M ) = ˆ ρ ⊕ ξ, and the associated projections: π ˆ ρ : ˆ ρ ⊕ ξ → ˆ ρ and π ξ : ˆ ρ ⊕ ξ → ξ. Define the two-form ˆ ω by ˆ ω = Ω ◦ ( π ξ × π ξ ) . (5)Here and throughout, we will also employ the following notation,ˆ X := X H ˆ λ ( X H ) . Lemma 2.5 (localization to a framed Hamiltonian energy pile) . Let ( f W , e Ω) be a symplectic manifold without boundary, and let e H : f W → R be a C ∞ -smooth proper Hamiltonian. Suppose further that zero is a regular value of e H .Then there exists an ǫ > , a framed Hamiltonian energy pile ( I ǫ × M, Ω , ˆ λ ) , anda C ∞ -smooth diffeomorphism Φ : I ǫ × M → {| e H | < ǫ } ⊂ f W for which ( e H ◦ Φ)( s, p ) = s and Φ ∗ e Ω = Ω . Additionally, the framed Hamiltonian energy pile can be found such that along theenergy level { } × M , the following are true(1) ˆ λ ( X H ) = 1 . (2) The vector bundles ˆ ρ and ξ are symplectic complements over { } × M .That is, for each q ∈ { } × M , each v q ∈ ˆ ρ q , and each w q ∈ ξ q , we have Ω( v q , w q ) = 0 . Proof.
We begin by fixing an auxiliary e Ω-compatible almost complex structure on f W ; denote it e J . This gives rise to the Riemannian metric g e J = e Ω ◦ (Id × e J ). Usingthis metric to compute the gradient of e H , we let ϕ t be the time t flow associated tothe vector field k∇ e H k − g e J ∇ e H in a neighborhood {| e H | < ǫ } for some small ǫ > M := e H − (0), it follows that for all sufficiently small ǫ > I ǫ × M → {| e H | < ǫ } ⊂ f W Φ( s, p ) = ϕ s ( p )is a diffeomorphism, and H := e H ◦ Φ satisfies H ( s, p ) = s .By construction, the vector field X e H is tangent to level sets of e H , and thus along { e H = 0 } we can define ˜ λ to be the one form uniquely determined by the conditions˜ λ ( X e H ) = 1 and ker ˜ λ = R ∇ e H ⊕ (cid:0) Span( X e H , ∇ e H ) (cid:1) ⊥ where ⊥ denotes the e Ω-symplectic complement. We then defineˆ λ = pr ∗ ˜ λ where pr : I ǫ × M → M is the canonical projection. It is straightforward to verifythat ˆ λ ( ∂ s ) = 0, and that L ∂ s ˆ λ = 0. Because ˆ λ ( X H ) (cid:12)(cid:12) { }× M = 1, it then followsthat ˆ λ ( X H ) > ǫ >
0. By construction then, ( W, Ω , ˆ λ ) is aframed Hamiltonian energy pile provided that ǫ > { } × M it is both the case that ˆ λ ( X H ) = 1 and the case ˆ ρ and ξ are Ω-symplectic complements. This completes the proof of Lemma 2.5. (cid:3) We make the following important observation.
Lemma 2.6 (energy levels are framed Hamiltonian manifolds) . Let (cid:0) I ǫ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω be theassociated structures defined above, see (3), (4), and (5). Then for each s ∈ I ǫ ,the restriction of ˆ λ and ˆ ω to the energy level { s } × M is a framed Hamiltonianstructure for this energy level.Proof. We begin by recalling that a framed Hamiltonian structure η = ( λ, ω ) foran odd-dimensional manifold M is a one-form λ and a closed two-form ω for which λ ∧ ω ∧ · · · ∧ ω =: vol M is a volume form. To show this latter non-degeneracycondition is satisfied on our energy levels, first note that on I ǫ × M we have that T ( I ǫ × M ) = ˆ ρ ⊕ ξ is a splitting and Ω is non-degenerate on each of ˆ ρ and ξ .Moreover, by construction ξ = ker ( ds ∧ ˆ λ ) and ˆ ρ = ker (ˆ ω ) = Span( ∂ s , X H ) , and ˆ ω = Ω ◦ ( π ξ × π ξ ) so that ds ∧ ˆ λ ∧ ˆ ω n > I ǫ × M ;here dim( M ) = 2 n + 1. It immediately follows thatˆ λ ∧ ˆ ω n > { s } × M for each s ∈ I ǫ . This establishes the non-degeneracy condition.
LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 13
Next, we establish that the restriction of ˆ ω to each energy level { s }× M is closed.To that end, let v, w ∈ T ( { s } × M ). Then there exist a, b ∈ R and v ξ , w ξ ∈ ξ suchthat v = aX H + v ξ and w = bX H + v ξ , and then Ω( v, w ) = Ω (cid:0) aX H + v ξ , bX H + w ξ (cid:1) = Ω( v ξ , w ξ )= Ω (cid:0) π ξ ( v ) , π ξ ( w ) (cid:1) = ˆ ω ( v, w )which shows that for the inclusion ι : { s } × M ֒ → I ǫ × M we have ˆ ω (cid:12)(cid:12) { s }× M = ι ∗ Ω,and thus the restriction of ˆ ω to { s } × M is closed. This completes the proof ofLemma 2.6. (cid:3) For the following considerations of adapted almost complex structures we sum-marize the salient points of the previous discussion. By definition a framed Hamil-tonian manifold ( ˇ M, ˇ λ, ˇ ω ) has an associated Reeb vector field ˇ X uniquely deter-mined by the equations ˇ λ ( ˇ X ) = 1 and i ˇ X ˇ ω = 0 . Given a framed Hamiltonian energy pile ( I ε × M, Ω , ˆ λ ) we can consider the Hamil-tonian H given by H ( s, m ) = s , which has Hamiltonian vector field X H is definedby i X Ω = − dH . We normalize it by setting b X = X H / ˆ λ ( X H ) . Then we observe that b X satisfiesˆ λ ( b X ) = 1 , ds ( b X ) = 0 , and i b X ˆ ω = 0 . Consequently, on each energy level { s } × M the Reeb vector field associated tothe framed Hamiltonian structure (ˆ λ, ˆ ω ) (cid:12)(cid:12) { s }× M is the restriction of b X .We now turn to equipping framed Hamiltonian energy piles with a certain classof almost complex structures, which we now define. Definition 2.7 (weakly adapted almost complex structure) . Let (cid:0) I ǫ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω be theassociated structures defined above. Let J be an almost complex structure on I ǫ × M which satisfies the following conditions.(J1) It preserves the splitting ˆ ρ ⊕ ξ ; that is, J : ˆ ρ → ˆ ρ, and J : ξ → ξ. (J2) There exists a smooth function of the form φ : I ǫ × M → (0 , φ ( s, p ) = φ ( s ) with the property that φ · J∂ s = b X where b X = X H ˆ λ ( X H ) . (J3) The following is a Riemannian metric g J ( v, w ) = ( ds ∧ ˆ λ + ˆ ω )( v, Jw ) . (6) In this case we say J is an almost complex structure weakly adapted to the framedHamiltonian energy pile ( I ǫ × M, Ω , ˆ λ ) . We pause for a moment to discuss the manner in which these almost complexstructures are weakly adapted. Consider a fixed framed Hamiltonian energy pile( I ε × M, Ω , ˆ λ ). Associated to this data we have the plane field bundles ξ = ker ( ds ∧ ˆ λ ) and ˆ ρ = ker (ˆ ω ) = Span( ∂ s , X H ) = Span( ∂ s , b X ) , with the two-form ˆ ω = Ω ◦ ( π ξ × π ξ ). Here we recall that T ( I ε × M ) = ˆ ρ ⊕ ξ. Following motivation from Symplectic Field Theory (SFT) as proposed in [6], thereis a somewhat natural choice of class of almost complex structures here, namelythose of the form J∂ s = b X and J : ξ → ξ where ˆ ω ◦ (Id × J ) (cid:12)(cid:12) ξ is symmetric and positive definite. There are two issues ofnote, which make such a choice somewhat different from the weakly adapted almostcomplex structures defined above. The first such difference is that in general ˆ ω isnot translation invariant, Put another way, this means that b X has s dependency,or rather that b X ( s, p ) = b X ( s ′ , p ) in general for s = s ′ . Because b X is not translationinvariant, and in fact in general the line bundle R b X ⊂ ˆ ρ will fail to be translationinvariant, it follows that any almost complex structure which preserves ˆ ρ will alsofail to be translation invariant.It is worth recalling that the framework of SFT typically requires a translationinvariant almost complex structure (in cylindrical homogeneous regions of symplec-tizations) or else is only required to be symplectically tame (in the inhomogenousor cobordant regions). In contrast, Definition 2.7 requires the almost complexstructure to be carefully “adapted” in an inhomogeneous region. Perhaps a moreimportant feature of Definition 2.7 is the “weakness” condition, which allows thealmost complex structure to have the form φ · J∂ s = b X . This is best understoodas a partial degeneration of the almost complex structure which can be undone bya finite amount of “stretching the neck” along the first factor of I ǫ × M . Moreprecisely, we define the embedding.Ψ : I ǫ × M → R × M (7) Ψ( s, p ) = (cid:0) ψ ( s ) , p (cid:1) where ψ ( s ) = Z s φ ( t ) dt. (8)To see the utility of the map Ψ, we consider an example, which already explainsits key features with respect to the other relevant data. Specifically, we considerthe case that φ ( s ) = δ where δ is some very small positive number, and we assume LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 15 that J ξ := J (cid:12)(cid:12) ξ : ξ → ξ is translation invariant in the s direction; that is, L ∂ s J ξ = 0.For v ξ ∈ ξ we then have J (cid:0) a∂ s + b b X + v ξ (cid:1) = − δb∂ s + aδ b X + J ξ v ξ . We also see that in this caseΨ : ( − ǫ, ǫ ) × M → (cid:16) − ǫδ , ǫδ (cid:17) × M (diffeomorphism)Ψ( s, p ) = ( sδ − , p )and thus (Ψ ∗ J ) (cid:0) a∂ ˇ s + b (Ψ ∗ b X ) + v ξ (cid:1) = − b∂ ˇ s + a (Ψ ∗ b X ) + J ξ v ξ , where ˇ s is the coordinate on ( − ǫδ − , ǫδ − ). Note that if we abuse notation byidentifying { s } × M with M , then(Ψ ∗ b X )(ˇ s, p ) = b X ( δ ˇ s, p ) . Similarly, if J ξ = J (cid:12)(cid:12) ξ has s dependence, then by abusing notation again, we have(Ψ ∗ J ξ )(ˇ s, p ) = J ξ ( δ ˇ s, p ) . Put another way, because the tangent bundle splits as ˆ ρ ⊕ ξ , and because the almostcomplex structures preserve this splitting, and the stretching direction is containedin ˆ ρ , it follows that this “neck stretching” does not degenerate the restriction J ξ ,and it is chosen to stretch the compressed J into something more well behaved. Forexample, (Ψ ∗ J ) ∂ ˇ s = Ψ ∗ b X . It is left to the reader to observe that this qualitativebehavior is preserved when one changes from the example case of φ ( s ) = δ to themore general case that φ : I ǫ × M → (0 ,
1] with φ ( s, p ) = φ ( s ).We will elaborate further on this stretching construction below, however it willbe helpful to put it in the context of “adiabatic degeneration”, which amounts tofully neck-stretching along a continuum of energy levels simultaneously. We makethis precise with the following. Definition 2.8 (adiabatically degenerating almost complex structures) . Let (cid:0) I ǫ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω bethe associated structures defined above, and let b X = X H / ˆ λ ( X H ) as above. Let J = { J k } k ∈ N be a sequence of weakly adapted almost complex structures in the senseof Definition 2.7, with φ k · J k ∂ s = b X , which also satisfies the following conditions.( J (Adiabatic Degeneration) We require that φ k → in C ∞ as k → ∞ .( J (Symplectic Area Controls Metric Area) For each k ∈ { , , , . . . } , and each v ∈ T ( I ǫ × M ) , we require that ( ds ∧ ˆ λ + ˆ ω )( v, J k v ) ≤ v, J k v ) ( J (Geometrically Bounded) There exists a sequence { C n } ∞ n =0 of positive con-stants and an auxiliary translation invariant metric ˜ g on R × M for which sup k ∈ N k (Ψ k ) ∗ J k k C n ≤ C n for each n ∈ N ; here k · k C n is the C n -norm on R × M with respect to theauxiliary metric ˜ g , and the Ψ k are the embeddings as in equation (7). We then say that J = { J , J , . . . } is a sequence of adiabatically degenerating almostcomplex structures adapted to the framed Hamiltonian energy pile given by ( I ǫ × M, Ω , ˆ λ ) . We note that given a framed Hamiltonian energy pile, it is easy to construct acandidate sequence of adiabatically degenerating almost complex structures. Forexample, one might fix a translation invariant J ξ = J (cid:12)(cid:12) ξ , and then define J k ∂ s = k b X and J k (cid:12)(cid:12) ξ = J ξ . In such a candidate, one can easily see that φ k = k − → k → ∞ , and infact (Ψ k ) ∗ J k is independent of k . Consequently ( J
1) and ( J
3) are immediatelysatisfied. However, ( J
2) is less obvious. That is, it is unclear if the followingestimate holds for all J k :( ds ∧ ˆ λ + ˆ ω )( v, J k v ) ≤ v, J k v )Geometrically, the concern is that while the almost complex structures J k are allΩ-tame, they are not Ω-compatible. Put another way, although the J k preservethe splitting T ( I ǫ × M ) = ˆ ρ ⊕ ξ , and ˆ ρ and ξ are each symplectic subspaces, thesesubspaces are not symplectic complements; that is there exist v ∈ ˆ ρ and w ∈ ξ suchthat Ω( v, w ) = 0. That the cross terms might cause an issue is then compoundedby the fact that we are degenerating adiabatically so that φ k → k → ∞ .We resolve this issue by making use of the fact that Lemma 2.5 allows us to guar-antee that along { } × M the sub-bundles ˆ ρ and ξ are Ω-symplectic complements.As it will turn out, condition ( J
2) can then be achieved by first fixing ǫ > C ∞ converging functions φ k . This is achieved via Proposition2.9 below. Proposition 2.9 (adiabatically degenerating constructions) . Let (cid:0) I ℓ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω be theassociated structures defined above, and assume, as in the conclusions of Lemma2.5, that along { } × M the sub-bundles ˆ ρ and ξ are symplectic complements and ˆ λ ( X H ) = 1 . Let b X = X H / ˆ λ ( X H ) as above. Let { ˇ J k } k ∈ N be a sequence of almostcomplex structures on I ℓ × M which satisfy the following conditions.(D1) ˇ J k : ˆ ρ → ˆ ρ and ˇ J k : ξ → ξ for each k ∈ N ,(D2) ( ds ∧ ˆ λ + ˆ ω ) ◦ (Id × ˇ J k ) is a Riemannian metric for each k ∈ N (D3) there exist constants { C ′ n } n ∈ N such that sup k ∈ N k ˇ J k k C n ≤ C ′ n Then there exist an ǫ > with the following significance. For any sequence offunctions φ k : ( − ǫ, ǫ ) → (0 , which converge in C ∞ (though not necessarily tozero), the weakly adapted almost complex structures defined by φ k · J k ∂ s = b X and J k (cid:12)(cid:12) ξ := ˇ J k (cid:12)(cid:12) ξ (9) satisfy the following properties(E1) For each v ∈ T ( I ǫ × M ) we have ( ds ∧ ˆ λ + ˆ ω )( v, J k v ) ≤ v, J k v ) . LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 17 (E2) There exists a sequence { K n } ∞ n =0 of positive constants and an auxiliary trans-lation invariant metric ˜ g on R × M for which sup k ∈ N k (Ψ k ) ∗ J k k C n ≤ K n for each n ∈ N ; here k · k C n is the C n -norm on R × M with respect to theauxiliary translation invariant metric ˜ g , and the Ψ k are the embeddings as inequation (7).In particular, if ǫ ′ ∈ (0 , ǫ ) and φ k (cid:12)(cid:12) ( − ǫ ′ ,ǫ ′ ) → , then the almost complex structures J k are adiabatically degenerating on I ǫ ′ × M in the sense of Definition 2.8, and on I ǫ × M these almost complex structures are all tame. The proof of Proposition 2.9 is elementary but somewhat lengthy, so we relegateit to Appendix A.1, however at present we indicate its utility. We start by notingthat a natural way to perform an adiabatic degeneration is to start with somefixed almost complex structure J which preserves the splitting ˆ ρ ⊕ ξ and for which( ds ∧ ˆ λ + ˆ ω ) ◦ (Id × J ) is a Riemannian metric. This almost complex structure willthen be Ω-compatible along { } × M , and hence Ω-tame in some neighborhood of { } × M . Consequently, J can be adjusted away from { } × M so that it is Ω-tameeverywhere, and still satisfies our compatibility conditions in a neighborhood of { } × M . Proposition 2.9 then guarantees that we can find φ k which tend to zeroin a neighborhood of zero so that the sequence of almost complex structures givenby φ k · J k ∂ s = b X and J k (cid:12)(cid:12) ξ = J (cid:12)(cid:12) ξ are everywhere tame, while also adiabatically degenerating a neighborhood of { }× M . Among other things, this means( ds ∧ ˆ λ + ˆ ω )( v, J k v ) ≤ v, J k v )for all k ∈ N . This latter condition is important, since by assumption the pseudo-holomorphic curves we will study will have uniform bounds on Ω-energy, which (asa consequence of the above inequality) will give us bounds on the area associatedto the metric ( ds ∧ ˆ λ + ˆ ω ) ◦ (Id × J ).With Proposition 2.9 stated and its use outlined, we now turn our attention toproving our main result. This is the topic of the next section.3. Proof of the Main Result
In this section we prove Theorem 2, which will be accomplished as follows. Inlight of Lemma 2.5 and Proposition 2.9, we see that the general problem can essen-tially be reduced to a localized problem involving framed Hamiltonian energy pilesand adiabatically degenerating almost complex structures. As such, our first taskis to prove Theorem 4 below, which is essentially a localized version of Theorem2. The proof of Theorem 4 is somewhat technical and will take the bulk of thissection. After this is established, Theorem 2 will follow rather quickly. We nowproceed with our first main technical argument.
Theorem 4 (localized almost existence) . Let (cid:0) I ǫ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω bethe associated structures. Let { J k } k ∈ N be a sequence of adiabatically degeneratingalmost complex structures in the sense of Definition 2.8. Suppose that for each k ∈ N there exists a proper pseudoholomorphic map u k : ( S k , j k ) → ( I ǫ × M, J k ) without boundary such that s ◦ u k ( S k ) = I ǫ , with the property that there exist noconnected components of S k on which u k is the constant map. Suppose further thatthere exist positive constants C g and C Ω for which Genus( S k ) ≤ C g and Z S k u ∗ k Ω ≤ C Ω . Then for almost every point s ∈ I ǫ , there exists a periodic orbit of the Hamiltonianvector field X H on the energy level { s } × M . That is, the set I ′ ǫ ⊂ I ǫ of energylevels of H ( s, p ) = s which contain a Hamiltonian periodic orbit has full measure: µ ( I ′ ǫ ) = µ ( I ǫ ) = 2 ǫ. In order to begin, we will first need to recall a version of the co-area formula asfollows.
Proposition 3.1 (The co-area formula) . Let ( S, g ) be an oriented C -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 (10) Z S f k∇ β k g dµ g = Z ba (cid:16) Z β − ( t ) f dµ g (cid:17) dt where ∇ β is the gradient of β computed with respect to the metric g .Proof. This is a well known result, however the details of this specific version areprovided in Appendix A.2 of [8]. (cid:3)
The co-area formula above will be used in a rather particular way, namely as ameans of expressing R S u ∗ ( ds ∧ ˆ λ ) as a double integral. More precisely, we have thefollowing. Lemma 3.2 (co-area application) . Let (cid:0) I ǫ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let J be a weaklyadapted almost complex structure in the sense of Definition 2.7. Let ( u, S, j ) be a J -holomorphic curve in I ǫ × M for which ∂S = ∅ . Then(11) Z S u ∗ ( ds ∧ ˆ λ ) = Z I ǫ (cid:16) Z ( s ◦ u ) − ( t ) \X u ∗ ˆ λ (cid:17) dt, where X := { ζ ∈ S : d ( s ◦ u ) ζ = 0 } ; that is, X is the set of critical points of thefunction s ◦ u : S → I ǫ ⊂ R .Proof. Define e S := S \ X , which is a manifold since e S ⊂ S is open. Observe thatby definition of X it follows that u ∗ ( ds ∧ ˆ λ ) (cid:12)(cid:12) X ≡
0, so Z S u ∗ ( ds ∧ ˆ λ ) = Z e S u ∗ ( ds ∧ ˆ λ ) . Since u : e S → I ǫ × M is an immersion, we may define the metric γ = u ∗ g J where g J is the Riemannian metric as in equation (6) in Definition 2.7; note that J is a That is, there may exist Cauchy sequences, with respect to g , which do not converge in S . LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 19 g J -isometry. The almost complex structure j on S induces an orientation on e S ,and hence we have(12) Z e S u ∗ ( ds ∧ ˆ λ ) = Z e S u ∗ ( ds ∧ ˆ λ (cid:1) ( ν, τ ) dµ γ , where ( ν, τ ) is a positively oriented γ -orthonormal frame field, and dµ γ is the volumeform on e S associated to the metric γ . This observation is elementary, howeverdetails are provided in Appendix A.2 of [8]. Note that equation (12) holds forarbitrary orthonormal frame field ( ν, τ ), however we shall henceforth make use ofthe following particular frame.(13) ν := ∇ ( s ◦ u ) k∇ ( s ◦ u ) k γ and τ := jν. Because u : S → I ǫ is a J -holomorphic map and J is a g J -isometry, it follows that j is a γ -isometry. Also note that for each v ξ ∈ ξ and a, b ∈ R , we have J (cid:0) a∂ s + b b X + v ξ (cid:1) = aφ b X − φb∂ s + Jv ξ with Jv ξ ∈ ξ , and hence ds ◦ J = − φ ˆ λ and ˆ λ ◦ J = 1 φ ds. With ν and τ as in equation (13), it is then straightforward to verify the following.0 = ( u ∗ ˆ λ )( ν ) = u ∗ ds ( τ )0 < u ∗ (cid:0) φ ds (cid:1) ( ν ) = ( u ∗ ˆ λ )( τ )1 = k τ k γ = k ν k γ Also,(14) k∇ ( s ◦ u ) k γ = sup x ∈ T ζ S k x k γ =1 d ( s ◦ u )( x ) = sup x ∈ T ζ S k x k γ =1 ds ( T u · x ) = u ∗ ds ( ν ) , and(15) ( φ ◦ u ) k u ∗ ˆ λ k γ = u ∗ ( φ ˆ λ )( τ ) = − u ∗ ds ( jjν ) = k∇ ( s ◦ u ) k γ . With ( ν, τ ) defined as such, we have the following. Z e S u ∗ ( ds ∧ ˆ λ )( ν, τ ) dµ γ = Z e S ds ( T u · ν )ˆ λ ( T u · τ ) dµ γ = Z e S φ ◦ u k∇ ( s ◦ u ) k γ dµ γ We employ Lemma 3.2 on e S with β = s ◦ u , and f = φ ◦ u k∇ ( s ◦ u ) k γ to obtain Z e S φ ◦ u k∇ ( s ◦ u ) k γ dµ γ = Z I ǫ (cid:16) Z ( s ◦ u ) − ( t ) \X φ ◦ u k∇ ( s ◦ u ) k γ dµ γ (cid:17) dt = Z I ǫ (cid:16) Z ( s ◦ u ) − ( t ) \X ( u ∗ ˆ λ )( τ ) dµ γ (cid:17) dt = Z I ǫ (cid:16) Z ( s ◦ u ) − ( t ) \X u ∗ ˆ λ (cid:17) dt, and hence by combining equalities we have Z S u ∗ ( ds ∧ ˆ λ ) = Z I ǫ (cid:16) Z ( s ◦ u ) − ( t ) \X u ∗ ˆ λ (cid:17) dt, which establishes equation (11). (cid:3) With these preliminaries established, our next main task is to carefully passto a certain subsequence of our almost complex structures and pseudoholomorphiccurves. To that end, we first recall that φ k · J k ∂ s = b X = X H / ˆ λ ( X H ) where { φ k } k ∈ N is a sequence of positive functions converging to zero in C ∞ . Next we define thefollowing intervals. For each t ∈ I ǫ , and for each sufficiently large k ∈ N define theopen interval I ( t, k ) = ( t , t ) = { a ∈ R : t < a < t } where Z t t φ k ( s ) ds = 1 = Z tt φ k ( s ) ds. Introduce the map Sh c : R × M → R × M Sh c ( s, p ) = ( s − c, p ) . Lemma 3.3 (the Ψ tk are diffeomorphisms) . We introduce the maps Ψ tk defined by Ψ tk = Sh ψ k ( t ) ◦ Ψ k (16) where Ψ k and ψ k are respectively the maps given in equations (7) and (8) . Then Ψ tk induces a diffeomorphism Ψ tk : I ( t, k ) × M → ( − , × M. of the form ( s, p ) → ( g k ( s ) , p ) with Ψ tk ( t, p ) = (0 , p ) .Proof. By definitionΨ k ( s, p ) = (cid:18)Z s φ k ( τ ) dτ, p (cid:19) = (cid:18)Z t φ k ( τ ) dτ + Z st φ k ( τ ) dτ, p (cid:19) . Hence Ψ k ( s, p ) = (cid:18) ψ k ( t ) + Z st φ k ( τ ) dτ, p (cid:19) which implies Ψ tk ( s, p ) = Sh ψ k ( t ) ◦ Ψ k ( s, p ) = (cid:18)Z st φ k ( τ ) dτ, p (cid:19) . It follows immediately that Ψ tk has image ( − , × M and is a diffeomorphism. (cid:3) As we have seen, when k is large, φ k is close to zero, and Ψ k then “stretchesthe neck” to undo the partial degeneration done by φ k (via the condition that φ k · J k ∂ s = b X ); the shift function, Sh, then R -shifts in the target to put the imageof t ∈ I ( t, k ) at 0 ∈ ( − , I ( t, k ) is an open neighborhoodof t ∈ I ǫ , which when neck-stretched and shifted becomes the standard interval( − , LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 21
Under the hypotheses of Theorem 4 we are given a sequence of pseudoholomor-phic curves u k : ( S k , j k ) → ( I ε × M, J k )where we have a genus bound C g and a symplectic bound: g ( S k ) ≤ C g and Z S k u ∗ k Ω ≤ C Ω . (17)Note that we obtain from this, because each of the two-forms ds ∧ ˆ λ and ˆ ω evaluatenon-negatively on J k -complex lines, the following estimate: R S k u ∗ k ( ds ∧ ˆ λ ) + R S k u ∗ k ˆ ω = R S k u ∗ k ( ds ∧ ˆ λ + ˆ ω ) ≤ R S k u ∗ k Ω ≤ C Ω . (18)Note that the first inequality follows from the assumption that the J k are adiabat-ically degenerating; (see Definition 2.8 and more specifically condition J C Ω := 2 · C Ω so that Z S k u ∗ k ˆ ω ≤ ¯ C Ω and Z S k u ∗ k ( ds ∧ ˆ λ ) ≤ ¯ C Ω . (19)Our goal is the rather careful construction of subsequences with a certain numberof good properties. To that end, it will be helpful to recall that if { x k } k ∈ N is asequence of points, then a subsequence can be specified using a strictly increasingfunction f : N → N by writing { x f ( k ) } k ∈ N . A further subsequence can be definedusing a strictly increasing function h : N → f ( N ) giving { x h ( k ) } k ∈ N . The followingresult is at the heart of our further constructions. We shall write N for the unionof N and { } . Proposition 3.4 (the key inductive construction) . Given a sequence of pseudoholomorphic curves u k : ( S k , j k ) → ( I ε × M, J k ) satisfying the bound (17) there exists a sequence ( f m ) m ∈ N of strictly increasingmaps f m : N → N and a sequence ( L m ) m ∈ N of finite sets L m ⊂ cl( I ε ) with thefollowing properties. (1) L = ∅ and f ( k ) = k for all k ∈ N . (2) L m − ⊂ L m for m ∈ N (3) f m : N → f m − ( N ) and L m − ⊂ L m for m ∈ N . (4) For m ∈ N the following holds. Given t ∈ L m ∩ I ε there exists a sequence ( τ k ) ⊂ I ε converging to t such that the following limit exists and satisfiesthe inequality lim k →∞ Z ( s ◦ u fm ( k ) ) − ( I ( τ k ,f m ( k ))) u ∗ f m ( k ) ˆ ω > ¯ C Ω m . (5) For m ∈ N and given t ∈ I ε \ L m there exists no sequence τ k → t suchthat lim sup k →∞ Z ( s ◦ u fm ( k ) ) − ( I ( τ k ,f m ( k ))) u ∗ f m ( k ) ˆ ω > ¯ C Ω m Proof.
We now begin an inductive process of constructing a sequence of nestedsubsequences. The start of the inductive construction is obvious. We define L := ∅ and f : N → N by f ( k ) = k for k ∈ N . Then Item (1) holds and for m = 0 alsoItem (4) is trivially satisfied since L ∩ I ε = ∅ . Again for m = 0 we see that I ε \ L = I ε . If take an element t ∈ I ε we see that the symplectic bound (19)implies that there is no sequence τ k → t with the property (5). The statements ofItems (2) and (3) are only relevant for m ≥
1. Hence, with the choices we havemade all relevant statements hold for m = 0.Let us assume that for some m ∈ N we have carried out the constructions of L i and f i for i = 0 , ..., m − i = 1 , ..., m −
1. We shall now construct f m and L m . Specifically we note thatthe construction will be made so that f m : N → f m − ( N ) and L m − ⊂ L m . Since ¯ C Ω / m < ¯ C Ω / m − it follows from Item (4) (for the case m −
1) that forevery t ∈ L m − ∩ I ε there exists a sequence τ k → t such that the following limitexists and satisfies the given inequalitylim k →∞ Z ( s ◦ u fm − k ) ) − ( I ( τ k ,f m − ( k ))) u ∗ f m − ( k ) ˆ ω > ¯ C Ω m . Of course, this remains true if we pass to a subsequence. By the inductive con-struction we know that for a given t ∈ I ε \ L m − there does not exist a sequence τ k → t with the propertylim sup k →∞ Z ( s ◦ u fm − k ) ) − ( I ( τ k ,f m − ( k ))) u ∗ f m − ( k ) ˆ ω > ¯ C Ω m − . (20)However, there might be for such a t a sequence if we replace the right-hand sideby the smaller number ¯ C Ω m . Thus we consider the possible cases. Assume first thatfor every t ∈ I ε \ L m − there is no sequence for whichlim sup k →∞ Z ( s ◦ u fm − k ) ) − ( I ( τ k ,f m − ( k ))) u ∗ f m − ( k ) ˆ ω > ¯ C Ω m (21)holds. In this case we define L m := L m − and f m ( k ) = f m − ( k ). Then f m : N → f m − ( N ) and L m − ⊂ L m . Moreover Items (2)-(5) hold for m and the constructionfor the m -case is complete.Assume next we find a t m, ∈ I ε \ L m − for which we have a sequence τ k → t m, for which (21) holds. Then we take a subsequence h m, : N → f m − ( N ) for whichlim k →∞ Z ( s ◦ u hm, k ) ) − ( I ( τ hm, k ) ,h m, ( k ))) u ∗ h m, ( k ) ˆ ω > ¯ C Ω m . (22)Next we can ask if we find a t m, ∈ I ε \ ( L m − ∪ { t m, } ) for which we have asequence τ k → t m, such thatlim sup k →∞ Z ( s ◦ u hm, k ) ) − ( I ( τ k ,h m, ( k ))) u ∗ h m, ( k ) ˆ ω > ¯ C Ω m If that is not the case we define f m ( k ) = h m, ( k ) and L m = L m − ∪ { t m, } . Oneeasily verifies (2)-(5) and the construction for the m -case is complete.Otherwise we find a t m, ∈ I ε \ ( L m − ∪ { t m, } ) with the above mentionedproperty and we can take h m, : N → h m, ( N ) so that the left-hand limit existsand is greater than ¯ C Ω / m . Again we ask if we find a third point t m, ∈ L m − ∪{ t m, , t m, } with the same property. If that is not the case we define L m = L m − ∪ LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 23 { t m, , t m, } and f m ( k ) = h m, ( k ). Again one verifies Items (2)-(5). Otherwise weobtain a sequence τ k → t m, and h m, : N → h m, ( N ) such thatlim sup k →∞ Z ( s ◦ u hm, k ) ) − ( I ( τ k ,h m, ( k ))) u ∗ h m, ( k ) ˆ ω > ¯ C Ω m . In the outline above we consider several cases and in some of them the procedureterminates after a finite number of steps. The only possible way of the procedurenot to stop is that we find more and more points t m,n , n ∈ N , with the previouslydescribed properties. However, we claim that this procedure terminates after afinite number of steps n m so that we can define L m = L m − ∪ { t m, , .., t m,n m } and f m ( k ) = h m,n m ( k ) which satisfies by construction Items (2)-(5). To see that theprocedure terminates, assume otherwise. Pick a positive integer N such that N · − m > N different points t m, , .., t m,N ∈ I ε \ L m − . By construction wehave the maps h m, : N → f m − ( N ) ,h m, : N → h m, ( N ) , ... h m,N : N → h m,N − ( N ) , with the properties that for t m,n , n ∈ { , ..., N } , there exists a sequence τ nk → t m,n such that lim k →∞ Z ( s ◦ u hm,j ( k ) ) − ( I ( τ k ,h m,j ( k ))) u ∗ h m,j ( k ) ˆ ω > ¯ C Ω m . (23)Using (23) we obtain with H ( k ) := h m,N ( k ) and the sequences τ nk → t n for n =1 , ..., N ¯ C Ω ≥ lim sup k →∞ Z S k u ∗ H ( k ) ˆ ω ≥ N X n =1 lim k →∞ Z ( s ◦ u H ( k ) ) − ( I ( τ nk ,H ( k ))) u ∗ H ( k ) ˆ ω ≥ N · − m · ¯ C Ω > ¯ C Ω . Observe that we use that the sets I ( τ nk , H ( k )) are mutually disjoint for large k .Indeed, since τ nk → t m,n as k → ∞ and the points t m,n are mutually disjoint, thisfollows from the fact that the diameter of the intervals I ( τ nk , H ( k )) is shrinking to0. Thus we have shown that for each fixed m , the procedure which generates theset { t m, , t m, , . . . } terminates after a finite number of iterations, and hence this setis finite. This completes the proof of Proposition 3.4. (cid:3) With these subsequences established, we now pass to a diagonal subsequence bydefining k m := f m ( m ) for m ∈ N . (24) We also define the countable subset L := [ m ∈ N L m . (25)The following result is then an immediate consequence of the above construction. Lemma 3.5 (vanishing horizontal area) . Let the ( I ǫ × M, Ω , ˆ λ ) , { J k } k ∈ N , and u k : ( S k , j k ) → ( I ǫ × M, J k ) be as above, andlet { k m } m ∈ N be the subsequence given in equation (24). Then for each t ∈ I ǫ \ L ,we have lim m →∞ Z ( s ◦ u km ) − (cid:0) I ( t ,k m ) (cid:1) u ∗ k m ˆ ω = 0 . Proof.
Since t ∈ I ε \ L it holds that t L m for every m ∈ N . Hence there doesnot exist a sequence τ k → t withlim sup k →∞ Z ( s ◦ u fm ( k ) ) − ( I ( τ k ,f m ( k ))) u ∗ f m ( k ) ˆ ω > ¯ C Ω m and specifically we must havelim sup k →∞ Z ( s ◦ u fm ( k ) ) − ( I ( t ,f m ( k ))) u ∗ f m ( k ) ˆ ω ≤ ¯ C Ω m . (26)Since k m := f m ( m ) is the diagonal sequence we deduce from (26) that for every m ∈ N lim sup i →∞ Z ( s ◦ u ki ) − ( I ( t ,k i )) u ∗ k i ˆ ω ≤ ¯ C Ω m . (27)This implies the assertionlim m →∞ Z ( s ◦ u km ) − (cid:0) I ( t ,k m ) (cid:1) u ∗ k m ˆ ω = 0 . (cid:3) We will next make use of this subsequence, and with it we will be interested inthe ˆ λ integrals of our curves along various levels { t } × M . This is made precise inequation (29) below, however for the moment we note that we will be interested invarious properties of these functions, (e.g. that they are measurable and integrable).Indeed, studying properties and limits of such functions will ultimately yield thedesired result regarding existence of periodic orbits on almost every energy level.We now proceed with this argument.For each m ∈ N we define Y m to be the set of critical values of the functions s ◦ u k m : S k m → I ǫ ⊂ R . By Sard’s theorem, each of these sets has measure zero;that is, µ ( Y m ) = 0 for each m ∈ N . By countable sub-additivity, we then also have µ ( Y ) = 0 where Y := [ m ∈ N Y m . (28)For each m ∈ N we then define the functions F m : I ǫ → [0 , ∞ ) ⊂ R (29) F m ( t ) = (R ( s ◦ u km ) − ( t ) u ∗ k m ˆ λ if t ∈ I ǫ \ Y . LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 25
We now claim the following.
Lemma 3.6 ( F m is measurable) . For each m ∈ N , the function F m defined above is a measurable function. Moreover,for each m ∈ N , the function F m agrees with the function t Z ( s ◦ u km ) − ( t ) \X m u ∗ k m ˆ λ (30) almost everywhere; here X m = { ζ ∈ S : d ( s ◦ u k m ) ζ = 0 } as in Lemma 3.2.Proof. First fix m ∈ N and define e F m : I ǫ → [0 , ∞ ) ⊂ R e F m ( t ) = (R ( s ◦ u km ) − ( t ) u ∗ k m ˆ λ if t ∈ I \ Y m . Observe that F m and e F m agree almost everywhere and consequently if e F m is mea-surable so is F m . Because s ◦ u k m ( X m ) = Y m which has measure zero, it followsthat e F m is almost everywhere equals the function defined in equation (30). Thisestablishes the second part of the lemma.To establish the first part of the lemma it is sufficient to show that e F m ismeasurable. That is, it is sufficient to show that for each r ∈ [0 , ∞ ), the set e F − m (cid:0) [0 , r ) (cid:1) is measurable. To that end, note that by assumption in Theorem 4 wehave s ◦ u k m ( S k m ) = I ǫ and consequently for each s ∈ I ǫ \ Y m we have e F m ( s ) = Z ( s ◦ u km ) − ( s ) u ∗ k m ˆ λ > . It also follows that e F − m (0) = Y m . Note that Y m is closed in I ǫ . Also note that e F m (cid:12)(cid:12) I ǫ \Y m is differentiable, and hence continuous, and thus A r := (cid:0) e F m (cid:12)(cid:12) I ǫ \Y m (cid:1) − (cid:0) [0 , r ) (cid:1) is open in I ǫ \ Y m . That is, there exists an open set O ⊂ I ǫ such that A r = O ∩ ( I ǫ \ Y m ). However I ǫ \ Y m is open in I ǫ and hence A r is open in I ǫ . Consequently A r is measurable.However, we then have e F − m (cid:0) [0 , r ) (cid:1) = (cid:0) e F m (cid:12)(cid:12) I ǫ \Y m (cid:1) − (cid:0) [0 , r ) (cid:1) [ (cid:0) e F m (cid:12)(cid:12) Y m (cid:1) − (cid:0) [0 , r ) (cid:1) = A r ∪ Y m , with A r open (and hence measurable) and Y m having measure zero (and henceis measurable). We conclude that e F − m (cid:0) [0 , r ) (cid:1) is measurable, which completes theproof of Lemma 3.6. (cid:3) For the following discussion we introduce the map F which is defined as follows: F : I ε → R + ∪ {∞} F ( s ) = lim inf m →∞ F m ( s ) . (31)Then by standard measure theory results F is an extended measurable function,see [3]. With this definition in place the next guiding observation (made rigor-ous below) is that two important results hold. The first one is given in the nextproposition. Proposition 3.7 ( F is almost everywhere finite) . With F as just defined in (31) it holds thatmeasure ( { s ∈ I ǫ : F ( s ) = ∞} ) = 0 Proof.
In view of (11) we have the formula Z S km u ∗ k m ( ds ∧ ˆ λ ) = Z I ε Z ( s ◦ u kk ) − ( t ) \X u ∗ k m ˆ λ ! ds. Using (18) and (19) we infer that2 · C Ω = ¯ C Ω ≥ Z S km u ∗ k m ( ds ∧ ˆ λ ) = Z I ε F m ( s ) ds. In view of Fatou’s Lemma, recalling that F = lim inf F m , we obtain¯ C Ω ≥ lim inf m →∞ Z I ε F m ( s ) ds ≥ Z I ε F ( s ) ds. This shows that { s ∈ I ǫ : F ( s ) = ∞} has vanishing measure. (cid:3) The second result, which is more substantial, studies the points s satisfying F ( s ) < ∞ . We will establish that there is a periodic orbit on { s } × M provided F ( s ) < ∞ and s ∈ I ε \ ( L ∪ Y ). The proof of Theorem 4 then follows immediatelysince we have just established that { s ∈ I ǫ : F ( s ) < ∞} has (full measure) 2 ε and L ∪ Y has measure zero. Proposition 3.8 (bounded F yields periodic orbit) . Let ( I ǫ × M, Ω , ˆ λ ) , { J k } k ∈ N , and u k : ( S k , j k ) → ( I ǫ × M, J k ) be as in the hypothesesof Theorem 4. Let L be defined as in equation (25), Y be as defined in equation(28), and let F m be the functions defined in equation (29). Also, let s ∈ I ǫ \ ( L ∪Y ) ,and suppose F ( s ) = lim inf m →∞ F m ( s ) = lim N →∞ inf m ≥ N F m ( s ) < ∞ . Then there exists a periodic orbit on the energy level { s } × M .Proof. We begin by passing to a subsequence (still denoted with subscripts m ) sothat lim N →∞ inf m ≥ N F m ( s ) = lim m →∞ F m ( s ) = lim m →∞ Z ( s ◦ u km ) − ( s ) u ∗ k m ˆ λ =: C ˆ λ < ∞ . Let Ψ s k m : I ǫ × M → R × M be the embedding provided in equation (16), whoserestriction e Ψ m to I ( s , k m ) × M given by e Ψ m : I ( s , k m ) × M → ( − , × M e Ψ m = Ψ s k m (cid:12)(cid:12) I ( s ,k m ) × M is a diffeomorphism. The key feature is that e Ψ m maps s to 0 and stretches outthe shrinking intervals to length two. Define e S m := ( s ◦ u k m ) − (cid:0) I ( s , k m ) (cid:1) and ˜ j m := j k m (cid:12)(cid:12) e S m , LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 27 and ˜ u m : e S m → ( − , × M ˜ u m = e Ψ m ◦ u k m . Also, with a the coordinate on ( − , − , × M e J m := ( e Ψ m ) ∗ J k m , ˜ λ := ( e Ψ m ) ∗ ˆ λ (Note that there is no m -dependence on the left-hand side!)˜ ω m := ( e Ψ m ) ∗ ˆ ω ˜ g m := ( da ∧ ˜ λ + ˜ ω m ) ◦ (Id × e J m ) = da + ˜ λ + ˜ ω m ◦ (Id × e J m )We now make several observations which follow immediately from our construction.(1) The maps ˜ u m : ( e S m , ˜ j m ) → (cid:0) ( − , × M, e J m (cid:1) are proper pseudoholomor-phic maps without boundary and which lack constant components.(2) Genus( e S m ) ≤ C g (3) ( a ◦ ˜ u m ) − (0) = ∅ (4) R ( a ◦ ˜ u m ) − (0) ˜ u ∗ m ˜ λ → C ˆ λ < ∞ (5) ( − , × M , equipped with any of the (˜ λ, ˜ ω m ), is a realized Hamiltonian ho-motopy in the sense of Definition 2.9 of [8], with adapted almost Hermitianstructures ( e J m , ˜ g m ), recalled in Definition A.3 in the Appendix.(6) ˜ ω m → ˜ ω s in C ∞ as m → ∞ , where˜ ω s = pr ∗ s ˆ ω and pr s ( s, p ) = ( s , p ) . (7) sup m ∈ N k e J m k C k < ∞ and sup m ∈ N k ˜ g m k C k < ∞ for each k ∈ N .Note that as a consequence of the above, together with Theorem 8 in [8] (areabounds in a realized Hamiltonian homotopy), it follows that there exists a number C A > m ∈ N we haveArea ˜ u ∗ m ˜ g m ( e S m ) = Z e S m ˜ u ∗ m ( da ∧ ˜ λ + ˜ ω m ) ≤ C A . We then pass to a subsequence (still denoted with subscripts m ) so that ( e J m , ˜ g m )converges in C ∞ . We then apply the main result from [7] (namely Target-localGromov compactness) to pass to a further subsequence (still denoted with sub-scripts m ) and find compact Riemann surfaces ( e Σ m , ˜ j m ) ⊂ ( e S m , ˜ j m ) with smoothboundary for which˜ u m ( e S m \ e Σ m ) ⊂ (cid:0) ( − , − ) ∪ ( , (cid:1) × M for all m ∈ N , (32)and for which the sequence ˜ u m : e Σ m → ( − , × M converges in a Gromov senseto a (nodal) limit pseudoholomorphic map ˜ u ∞ : ( e Σ ∞ , ˜ j ∞ ) → (cid:0) ( − , × M, e J ∞ (cid:1) ;here ( e Σ ∞ , ˜ j ∞ ) is compact and may have smooth boundary. Note that because( a ◦ ˜ u m ) − (0) = ∅ for all m ∈ N , it follows from equation (32) that e Σ ∞ = ∅ , andif ∂ e Σ ∞ = ∅ , then ˜ u ∞ ( ∂ e Σ ∞ ) ⊂ (( − , − ) ∪ ( , × M . Because s ∈ I ǫ \ ( L ∪ X )(specifically because s / ∈ L ), we must have Z e Σ m ˜ u ∗ m ˜ ω m → Z e Σ m ˜ u ∗ m ˜ ω m → Z e Σ ∞ ˜ u ∗∞ ˜ ω s so Z e Σ ∞ ˜ u ∗∞ ˜ ω s = 0 . We now make the following claim.
Lemma 3.9 (non-trivial limit component with boundary) . There exists a connected component e Σ ′ of e Σ ∞ for which e Σ ′ ∩ ( a ◦ ˜ u ∞ ) − (0) = ∅ and ∂ e Σ ′ = ∅ .Proof. Suppose not. For notational clarity, we let S denote the set of connectedcomponents of e Σ ∞ which have non-empty intersection with ( a ◦ ˜ u ∞ ) − (0). Thenthere are three possibilities. Case I: S = ∅ .In this case, e Σ ∞ can be written as the disjoint union e Σ ∞ = e Σ + ∞ ∪ e Σ −∞ where a ◦ ˜ u ∞ ( e Σ + ∞ ) ⊂ (0 ,
1) and a ◦ ˜ u ∞ ( e Σ −∞ ) ⊂ ( − , . If either of e Σ ±∞ are empty, then by Gromov convergence there must be some large m ∈ N and some number a ∈ ( − , ) for which ( a ◦ ˜ u m ) − ( a ) = ∅ , and hence thereexists a s ∈ I ǫ (specifically s = ψ − k m ( a )) for which ( s ◦ u k m ) − ( s ) = ∅ . However,this violates the assumption in Theorem 4 which states that s ◦ u k ( S k ) = I ǫ for all k . Consequently e Σ + ∞ = ∅ and e Σ −∞ = ∅ .Next note that because e Σ ∞ is compact, it follows that each of e Σ ±∞ are compactand non-empty. However, we then havesup z ∈ e Σ −∞ a ◦ ˜ u ∞ ( z ) = a − < < a + = inf z ∈ e Σ + ∞ a ◦ ˜ u ∞ ( z ) . But then again by Gromov convergence, this will violate the assumption that s ◦ u k ( S k ) = I ǫ for all k . Thus Case I is impossible. Case II:
For each e Σ ′ ∈ S , the restriction ˜ u ∞ (cid:12)(cid:12) e Σ ′ is the constant map.In this case, we can write e Σ ∞ as the disjoint union e Σ ∞ = e Σ + ∞ ∪ e Σ ∞ ∪ e Σ −∞ where a ◦ ˜ u ∞ ( e Σ + ∞ ) ⊂ (0 , , a ◦ ˜ u ∞ ( e Σ ∞ ) = { } , and a ◦ ˜ u ∞ ( e Σ −∞ ) ⊂ ( − , . The argument then proceeds as in Case I, which shows that Case II is also impos-sible.
Case III:
There exists e Σ ′ ∈ S , such that the restriction ˜ u ∞ (cid:12)(cid:12) e Σ ′ is not constant.Note that by the contradiction hypothesis, we must have ∂ e Σ ′ = ∅ , and by Gromovconvergence e Σ ′ is compact. That is, ( e Σ ′ , ˜ j ∞ ) is a closed Riemann surface. Next,we make use of the fact that Z e Σ ∞ ˜ u ∗∞ ˜ ω s = 0together with the fact that ˜ ω s evaluates non-negatively on e J ∞ -complex lines toconclude that for each z ∈ e Σ ′ we must haveImage( T z ˜ u ∞ ) ⊂ ker(˜ ω s ) ˜ u ∞ ( z ) = Span (cid:0) ∂ a , X (˜ u ∞ ( z )) (cid:1) , LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 29 where X is the Hamiltonian vector field X ( a, p ) = b X ( s , p ). Consequently, thereexists a Hamiltonian trajectory γ : R → { } × M , solving γ ′ ( t ) = X ( γ ( t )) for all t for which ˜ u ∞ ( e Σ ′ ) ⊂ ( − , × γ ( R ) . If γ is not periodic, then the mapΦ : ( − , × R → R × M Φ( s, t ) = (cid:0) s, γ ( t ) (cid:1) is an injective pseudoholomorphic immersion, and hence the mapΦ − ◦ ˜ u ∞ : e Σ ′ → ( − , × R ⊂ C is a holomorphic map from a closed Riemann surface into C . By the maximumprinciple, and the fact that Φ is an immersion, it follows that ˜ u ∞ : e Σ ′ → ( − , × M is a constant map, but this contradicts the assumption defining Case III.The case in which γ is a periodic orbit is treated similarly by holomorphicallyparameterizing ( − , × γ ( S ) by an annulus in C . Again the maximum principleapplies and we conclude that ˜ u ∞ : e Σ ′ → ( − , × M is a constant map which isimpossible. We conclude that Case III is impossible.All cases are impossible, and hence this completes the proof by contradiction ofLemma 3.9. (cid:3) With Lemma 3.9 established, we now observe that there exists a connectedcomponent e Σ ′ ⊂ e Σ ∞ for which e Σ ′ ∩ ( a ◦ ˜ u ∞ ) − (0) = ∅ and ∂ e Σ ′ = ∅ . We make useof the fact that ˜ ω s evaluates non-negatively on e J ∞ -complex lines, together withthe fact that Z e Σ ∞ ˜ u ∗∞ ˜ ω s = 0 , to conclude that ˜ u ∞ ( e Σ ′ ) ⊂ ( − , × γ ( R ) for some Hamiltonian trajectory γ : R → M . That is, ˙ γ = X ( γ ) where ˆ λ ( X ) = 1 and i X ˜ ω s = 0. If γ is not periodic, thenthe map Φ : ( − , × R → ( − , × M Φ( s, t ) = (cid:0) s, γ ( t ) (cid:1) is an injective pseudoholomorphic immersion, and hence the map v : e Σ ′ → ( − , × R ⊂ C v = Φ − ◦ ˜ u ∞ is a non-constant holomorphic map from a connected compact Riemann surface e Σ ′ with non-empty boundary into C . Moreover, this holomorphic map satisfies thefollowing two conditions:(1) v ( ∂ e Σ ′ ) ⊂ (cid:0) ( − , − ) ∪ ( , (cid:1) × R .(2) v − ( { } × R ) = ∅ .However, by the maximum principle, this is impossible. We conclude that γ mustbe a periodic trajectory of the Hamiltonian vector field X which satisfiesˆ λ ( X ) = 1 , and i X ˜ ω s = 0 . Or in other words, for the symplectic manifold ( I ǫ × M, Ω), and Hamiltonian func-tion H ( s, p ) = s , there exists a periodic Hamiltonian orbit on energy level { s }× M .This completes the proof of Proposition 3.8. (cid:3) Let us summarize the already established facts involving the map F : I ε → [0 , + ∞ ]. Recall that F has been given as F ( s ) := lim inf m →∞ F m , where the F m have been previously defined in (30) by F m : I ǫ → [0 , ∞ ) ⊂ R (33) F m ( t ) = (R ( s ◦ u km ) − ( t ) u ∗ k m ˆ λ if t ∈ I ǫ \ Y . The Y has been defined in (28) and we have shown the following. • The set { s ∈ I ε | F ( s ) = ∞} has measure zero. • The set L is countable and the set Y has measure zero. • If s ∈ {I ε | F ( s ) < ∞} \ ( L ∪ Y ) then { s } × M contains a periodic orbit.We are now prepared to prove the following. Theorem 2 (Main Result) . Let ( W, Ω) be a symplectic manifold without boundary, and let H : W → R be asmooth proper Hamiltonian. Fix E − , E + ∈ H ( W ) ⊂ R with E − < E + , as wellas positive constants, C g > , and C Ω > . Suppose that for each Ω -tame almostcomplex structure J on W there exists a proper pseudoholomorphic map u : ( S, j ) → { p ∈ W : E − < H ( p ) < E + } without boundary, which also satisfies the following conditions:(1) (genus and area bounds) The following inequalities hold:
Genus( S ) ≤ C g and Z S u ∗ Ω ≤ C Ω . (2) (energy surjectivity) The map H ◦ u : S → ( E − , E + ) is surjective.Then there is a periodic Hamiltonian orbit on almost every energy level in range ( E − , E + ) . That is, if we let I ⊂ ( E − , E + ) denote the energy levels of H whichcontain a Hamiltonian periodic orbit, then I has full measure: µ ( I ) = µ (cid:0) ( E − , E + ) (cid:1) = E + − E − . Proof.
In order to prove this result, we will make use of our localization results,but first we need to properly reframe the problem. For notational convenience webegin by defining: f W = { p ∈ W : E − < H ( p ) < E + } . Next, for each c ∈ R , we define the function H c : f W → R H c ( q ) = H ( q ) − c and observe that H and H c generate identical Hamiltonian vector fields on f W .Consequently, γ : R → f W is a Hamiltonian periodic obit of H if and only if it is a By proper here, we mean that for each compact set
K ⊂ R , the set H − ( K ) is compact. LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 31
Hamiltonian periodic orbit of H c . Next we make the following claim. Claim:
To prove Theorem 2, it is sufficient to show that for each regular value c ∈ ( E − , E + ) of H , there exists a δ = δ ( c ) > such that the set of energy levels {| H c | < δ } containing a Hamiltonian periodic orbit has measure δ . To see that this claim is true, we first consider the case that H has no criticalpoints in f W . In this case, every c ∈ ( E − , E + ) is a regular energy value, and thusfor each such c we define δ c = δ ( c ) so that the set of energy levels {| H c | < δ c } containing a Hamiltonian periodic orbit has measure 2 δ c .It follows that { ( c − δ c , c + δ c ) } E −
0, the aim will be to apply Proposition 2.9. However todo that we must first have at our disposal a sequence of suitable almost complexstructures. To construct these, we start by defining an almost complex J c on I ǫ c × M by requiring that J c ∂ s = b X c and that J c : ξ c → ξ c have the property that J ξ c := J c (cid:12)(cid:12) ξ c be translation invariant, and that ˆ ω c ◦ (Id × J ξ c ) is symmetric andpositive definite. We then treat J c as a constant sequence and apply Proposition2.9 which guarantees the existence of an ℓ c (stated in the proposition as ǫ ) withthe following significance. Let φ ck : I ℓ c → (0 ,
1] be a sequence of functions whichconverge in C ∞ to a limit function φ c ∞ which satisfies φ c ∞ ( s ) = ( | s | ≥ ℓ c | s | ≤ ℓ c . Then a consequence of Proposition 2.9 is that the almost complex structures definedby φ ck · J k ∂ s = b X c and J k (cid:12)(cid:12) ξ c = J ξ c are each Ω c -tame on I ℓ c × M , and they are adiabatically degenerating on I ℓ c / × M .We define δ c := ℓ c . We then observe that because these almost complex struc-tures are all tame, on I ℓ c × M they can be seen as arising as J k = (Φ c ) ∗ e J k for someΩ-tame almost complex structures e J k on f W . The hypotheses of Theorem 2 thenguarantee the existence of a sequence of pseudoholomorphic curves with boundedsymplectic area and genus, and which span the energy levels in I δ c × M , while thealmost complex structures are adiabatically degenerating on I δ c × M . Recall thatsecond countability of the domains of these pseudoholomorphic curves guaranteesthat the set of connected components on which each is a constant map is count-able, and hence for any such curve the set of energy levels containing a constantcomponent is countable. Making use of the fact that our pseudoholomorphic mapsare continuous and proper, it follows that one may remove the constant connectedcomponents while still guaranteeing “energy surjectivity.” Thus after removingconstant components, we may apply Theorem 4, which guarantees that the set ofenergy levels with periodic orbits has full measure in ( − δ c , δ c ). Because we havereduced the proof of Theorem 2 to establishing just this result, we see that we havecompleted the proof of Theorem 2. (cid:3) With Theorem 2 established, we now prove Theorem 3.
Theorem 3 (intertwining existence and almost existence) . Let ( W, Ω) be a four-dimensional compact connected exact symplectic manifold withboundary ∂W = M + ∪ M − . Suppose M + is positive contact type in the sense ofDefinition 1.6, and suppose that one of the following three conditions holds:(1) M + has a connected component diffeomorphic to S ,(2) there exists an embedded S ⊂ M + which is homotopically nontrivial in W ,(3) ( M + , λ ) has a connected component which is overtwisted.Then for each Hamiltonian H ∈ C ∞ ( W ) for which H − ( ±
1) = M ± , the followingis true. For each s ∈ [ − , the energy level H − ( s ) contains a closed non-empty setother than the energy level H − ( s ) itself which is invariant under the Hamiltonianflow of X H ; moreover for almost every s ∈ [ − , this closed invariant subset is aperiodic orbit. LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 33
Proof.
First observe that if s ∈ [ − ,
1] is a critical value of H , then there exists p ∈ H − ( s ) such that dH ( p ) = 0, and hence the constant trajectory γ : R → H − ( s ) γ ( t ) = p is periodic orbit. Also note that because M + = H − (1) is contact type, and iseither S , overtwisted, or contains a homotopically non-trivial S , it follows from[14] that M + has a periodic orbit. Then for any regular value s ∈ [ − , X H on H − ( s ) is not minimal.This establishes that each energy level H − ( s ) with s ∈ [ − ,
1] is not minimal. Tocomplete the proof of Theorem 3, it remains to show that almost every energy level H − ( s ) for s ∈ ( − ,
1) has a periodic orbit. This follows from Thereom 2 above,provided we can guarantee the existence of the required pseudoholomorphic curvesfor arbitrary tame almost complex structure. However, this is fairly standard andthe relevant details are provided in [8] (specifically the proof of Theorem 2), howeverwe recall the key points here.Choose an Ω-tame almost complex structure J on W , and fix regular values E − and E + of H so that − < E − < E + < . Consider the case that tight S is a connected component of M + . Then along thisspherical component, W can be symplectically capped off by C P \ O where O isdiffeomorphic to the four-ball, and the resulting symplectic manifold we denote by( f W , e Ω). Note that in order to guatantee that e Ω is indeed symplectic, one may needto require that its restriction to C P \ O ⊂ f W be a large constant multiple of theFubini-Study metric. By adjusting J in a neighborhood of S ⊂ M + , the almostcomplex structure J can be extended to an e Ω-tame almost complex structure e J on f W , which is also standard in a neighborhood of C P ⊂ C P \O . Define the constant C e Ω := R C P e Ω to be the e Ω-area of this sphere at infinity. One then considers theconnected component of the moduli space of e J -pseudoholomorphic curves whichcontain this C P ⊂ C P \ O ⊂ f W . As detailed in [8], automatic transversalityguarantees that this four real dimensional moduli space (of unparameterized curves)is cut out transversely, each distinct pair of curves intersects at exactly one point,and by requiring Ω to be exact there cannot be any “bubbles” that arise in thecompactification. Homotopy invariance of intersection numbers guarantees thateach curve in this moduli space intersects C P ⊂ C P \ O ⊂ f W . It remains toshow that this family of curves satisfies the energy surjectivity condition; that is,that this family of curves extends from M + = H − (1) to every energy level H − ( s )for s ∈ ( − , e J is e Ω-tame. This then guarantees energy surjectivity. That is, foreach Ω-tame almost complex structure J , there exists a pseudoholomorphic map˜ u : ( e S, j ) → ( f W , e J ) with the property that the restricted map u := ˜ u (cid:12)(cid:12) S where S := (cid:8) z ∈ e S : H ◦ ˜ u ( z ) ∈ ( E − , E + ) (cid:9) satisfies H ◦ u ( S ) = ( E − , E + ). It is also easily checked that genus( S ) = 0 and R S u ∗ e Ω ≤ C e Ω . Consequently, Theorem 2 applies, and hence almost every energy level in ( E − , E + ) contains a periodic orbit. By letting E − → − E + →
1, thedesired result is immediate.This covers the tight S case; the overtwisted case and the homotopically non-trivial S ⊂ M + case are very similar, although the mechanism to generate thecurves is different. The reader is directed to [8] for the details. (cid:3) Appendix A. Miscellaneous Support
This section is mostly devoted to providing a few support definitions and resultswhich are important but are otherwise a bit of a distraction from more importantarguments.A.1.
Supporting Proofs.
Our primary goal of this section is to prove Proposition2.9, however to do so we must first establish a few important supporting lemmata.The first of these is the following.
Lemma A.1 (cross term control) . Let (cid:0) I ℓ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ and ξ be the asso-ciated structures defined in Section 2, and assume as in the conclusions of Lemma2.5 that along { } × M , the sub-bundles ˆ ρ and ξ are Ω -symplectic complements.Let J be an Ω -compatible almost complex structure on I ℓ × M , with associatedRiemannian metric g = Ω ◦ (Id × J ) . Then for each δ > , there exists an ǫ = ǫ ( δ, Ω , ˆ λ, ˆ ω, J ) ∈ (0 , ℓ ) , with the property that for each v = v ˆ ρ + v ξ ∈ T ( I ǫ × M ) with v ˆ ρ ∈ ˆ ρ and v ξ ∈ ξ we have | Ω( v ˆ ρ , v ξ ) | ≤ δ k v ˆ ρ k g k v ξ k g . Proof.
Given s ∈ I ℓ we consider the embedding M → I ℓ × M : m → ( s, m )and the pull-backs of ˆ ρ → I ℓ × M and ξ → I ℓ × M , respectively. We denote themby ˆ ρ s and ξ s . Then the pull-back of the tangent space T ( I ℓ × M ) to M by thesame map has the direct sum decomposition ˆ ρ s ⊕ ξ s . In the case of s = 0 thisdecomposition is Ω-orthogonal. We define the functionΘ : ( − ℓ, ℓ ) → [0 , ∞ )Θ( s ) := sup v ∈ ˆ ρ s \{ } w ∈ ξ s \{ } | Ω( v, w ) |k v k g k w k g . It is straightforward to show that Θ is continuous and and that Θ(0) = 0. Bycompactness ˜Θ : [0 , ℓ ) → R ε → max s ∈ [ − ε,ε ] Θ( s )is also continuous and ˜Θ(0) = 0. The desired result is then immediate. (cid:3) In order to proceed further, we will need to define a certain metric bound on thegeometry of a weakly adapted almost complex structure. We denote this quantity |i J h| , and establish it as follows. Given a framed Hamiltonian energy pile( I ℓ × M, Ω , ˆ λ ) , LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 35 we first fix a background metric g on I ℓ × M associated to an auxiliary Ω-compatiblealmost complex structure J by the usual formula g := Ω ◦ (Id × J ). Of course,near the ends of I ℓ × M the metric g might not behave well. Our Hamiltonianenergy pile comes with the structures ˆ ω and ξ . Given any other weakly adaptedalmost complex structure J we define the following quantity, where ℓ ′ ∈ (0 , ℓ ): |i J h| ( I ℓ ′ × M, Ω , ˆ λ, ˆ ω,J ) := sup q ∈I ℓ ′ × Mv ξq ∈ ξ q \{ } max k v ξq k g J k v ξq k g , k v ξq k g k v ξq k g J , k Jv ξq k g k v ξq k g , k v ξq k g k Jv ξq k g ! ;(34)here g J is the metric as defined in equation (6) in Definition 2.7.An immediate benefit of such a definition is that whenever it is the case that |i J h| ≤ C , it is also the case that the following hold for each v ξ ∈ ξ lying above apoint in I ℓ ′ × M k v ξ k g ≤ C k v ξ k g J (35) k v ξ k g J ≤ C k v ξ k g (36) k Jv ξq k g ≤ C k v ξq k g (37) k v ξq k g ≤ C k Jv ξq k g . (38)It is also worth noting that if J and J are weakly adapted almost complex struc-tures which agree on ξ , then |i J h| = |i J h| . Indeed, the quantity |i J h| depends onlyon J (cid:12)(cid:12) ξ . With this definition established, we can now proceed with an importantapplication. Proposition A.2 (metric area controlled by symplectic area) . Let (cid:0) I ℓ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω bethe associated structures defined in equations (2), (3), (4), and assume, as in theconclusions of Lemma 2.5, that along { }× M the sub-bundles ˆ ρ and ξ are symplecticcomplements and ˆ λ ( X H ) = 1 . Let J be an auxiliary Ω -compatible almost complexstructure on I ℓ × M , with associated Riemannian metric g = Ω ◦ (Id × J ) . Fix alarge positive constant C > . Then there exists a number ǫ = ǫ (Ω , ˆ λ, ˆ ω, J ) ∈ (0 , ℓ ) with the property that after trimming I ℓ × M to I ǫ × M , the following holds. If J is a weakly adapted almost complex structure for which |i J h| ( I ǫ × M, Ω , ˆ λ, ˆ ω,J ) ≤ C in the sense of equation (34), then ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) ≤ v, Jv ) for each v ∈ T ( I ǫ × M ) . Here J only needs to be defined over J ε × M . The important fact is that ε does not depend on J and only on the numericalbound! Proof.
We first fix ℓ ′ ∈ (0 , ℓ ). This is done since the background metric g mightnot be well-behaved near the ends of I ℓ × M . We begin by defining the constants C and δ by C := sup q ∈I ℓ ′ × M k ∂ s k g and δ := 18 C C . (39) Let ǫ = ǫ ( δ, Ω , ˆ λ, ˆ ω, J ) be the constant guaranteed by Lemma A.1 and we mayassume without loss of generality that 0 < ε ≤ ℓ ′ . Recall that a consequence ofLemma A.1 is that for each v = v ˆ ρ + v ξ ∈ T ( I ǫ × M ) with v ˆ ρ ∈ ˆ ρ and v ξ ∈ ξ wehave | Ω( v ˆ ρ , v ξ ) | ≤ δ k v ˆ ρ k g k v ξ k g . (40)Also recall that along { } × M we have ˆ λ ( X H ) = 1, so that by shrinking ǫ >
0, wecan further guarantee that sup q ∈I ǫ × M (cid:12)(cid:12) ˆ λ ( X H ( q )) − (cid:12)(cid:12) ≤ . (41)Next recall that b X = X H ˆ λ ( X H ) , J∂ s = φ b X, and J b X = − φ∂ s . Additionally, recall that we have the splitting T ( I ǫ × M ) = ˆ ρ ⊕ ξ , and the associatedprojections π ˆ ρ : ˆ ρ ⊕ ξ → ˆ ρ and π ξ : ˆ ρ ⊕ ξ → ξ . In general we will use the abbreviatednotation v ˆ ρ = π ˆ ρ ( v ) and v ξ = π ξ ( v ). In the following we work with ˆ ρ ⊕ ξ → I ε × M ,where ε > ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) = ˆ λ ( X H ) · Ω( v ˆ ρ , Jv ˆ ρ ) for each v ˆ ρ ∈ ˆ ρ. (42)To prove this, we first write v ˆ ρ = a∂ s + b b X for some a, b ∈ R , and then compute asfollows. Ω( v ˆ ρ , Jv ˆ ρ ) = Ω (cid:0) a∂ s + b b X, J ( a∂ s + b b X ) (cid:1) = Ω (cid:0) a∂ s + b b X, φ a b X − φb∂ s (cid:1) = (cid:16) a φ + φb (cid:17) Ω( ∂ s , b X )= (cid:16) a φ + φb (cid:17) Ω( ∂ s , X H )ˆ λ ( X H )= (cid:16) a φ + φb (cid:17) dH ( ∂ s )ˆ λ ( X H )= (cid:16) a φ + φb (cid:17) ds ( ∂ s )ˆ λ ( X H )= (cid:16) a φ + φb (cid:17) λ ( X H ) . Similarly, we compute the following (with the same v ˆ ρ ∈ ˆ ρ as above).( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) = ( ds ∧ ˆ λ ) (cid:0) a∂ s + b b X, J ( a∂ s + b b X ) (cid:1) = ( ds ∧ ˆ λ ) (cid:0) a∂ s + b b X, φ a b X − φb∂ s (cid:1) = (cid:16) a φ + φb (cid:17) = ˆ λ ( X H ) Ω( v ˆ ρ , Jv ˆ ρ ) . LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 37
This establishes equation (42). The case with ˆ ω is much easier. Indeed, recall thatby definition we have ˆ ω = Ω ◦ ( π ξ × π ξ ). It immediately then follows thatˆ ω ( v ξ , Jv ξ ) = Ω( v ξ , Jv ξ ) for each v ξ ∈ ξ. (43)In just a moment we will be concerned with estimating cross terms, however firstwe will need an elementary estimate. Starting with k ∂ s k g J = ( ds ∧ ˆ λ )( ∂ s , J∂ s ) = ( ds ∧ ˆ λ )( ∂ s , φ b X ) = φ . and combining the above with the definition of C in equation (39) yields k ∂ s k g ≤ C = φC k ∂ s k g J ≤ C k ∂ s k g J . (44)We are now prepared to estimate cross terms. To that end, we let v = v ˆ ρ + v ξ ∈ T ( I ǫ × M ) with v ˆ ρ = a∂ s + b b X ∈ ˆ ρ and v ξ ∈ ξ . Then | Ω( v ˆ ρ , Jv ξ ) | = | Ω( a∂ s + b b X, Jv ξ ) | = (cid:12)(cid:12)(cid:12) Ω( a∂ s , Jv ξ ) + b Ω( X H , Jv ξ )ˆ λ ( X H ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Ω( a∂ s , Jv ξ ) − b dH ( Jv ξ )ˆ λ ( X H ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Ω( a∂ s , Jv ξ ) − b ds ( Jv ξ )ˆ λ ( X H ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Ω( a∂ s , Jv ξ ) (cid:12)(cid:12)(cid:12) by equation (4), and J : ξ → ξ ≤ δ k a∂ s k g k Jv ξ k g by equation (40) ≤ δC k a∂ s k g k v ξ k g by equation (37) ≤ δC k a∂ s k g k v ξ k g J by equation (35) ≤ δC C k a∂ s k g J k v ξ k g J by equation (44)= k a∂ s k g J k v ξ k g J by equation (39) ≤ (cid:0) k a∂ s k g J + k v ξ k g J (cid:1) ≤ (cid:0) k a∂ s k g J + k b b X k g J + k v ξ k g J (cid:1) ≤ (cid:0) k v ˆ ρ k g J + k v ξ k g J (cid:1) = k v k g J = ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) . The other cross term is estimated rather similarly. Again let v = v ˆ ρ + v ξ ∈ T ( I ǫ × M )with v ˆ ρ = a∂ s + b b X ∈ ˆ ρ and let v ξ ∈ ξ . We estimate: | Ω( v ξ , Jv ˆ ρ ) | = (cid:12)(cid:12) Ω (cid:0) v ξ , J ( a∂ s + b b X ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) Ω( v ξ , aφ b X − bφ∂ s ) (cid:12)(cid:12) = (cid:12)(cid:12) Ω( v ξ , bφ∂ s ) (cid:12)(cid:12) ≤ δ k v ξ k g k bφ∂ s k g ≤ δC C φ k v ξ k g J k b∂ s k g J ≤ k v ξ k g J k b∂ s k g J ≤ (cid:0) k v ξ k g J + k b∂ s k g J ) ≤ (cid:0) k v ξ k g J + k v ˆ ρ k g J ) ≤ k v k g J = ( ds ∧ ˆ λ + ˆ ω )( v, Jv )With these two estimates established, we can now use them to establish the follow-ing. (cid:12)(cid:12) Ω( v, Jv ) − ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) (cid:12)(cid:12) = (cid:12)(cid:12) Ω (cid:0) v ˆ ρ + v ξ , J ( v ˆ ρ + v ξ ) (cid:1) − ( ds ∧ ˆ λ + ˆ ω ) (cid:0) v ˆ ρ + v ξ , J ( v ˆ ρ + v ξ ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) Ω (cid:0) v ˆ ρ + v ξ , J ( v ˆ ρ + v ξ ) (cid:1) − ( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) − ˆ ω ( v ξ , Jv ξ ) (cid:12)(cid:12) ≤ | Ω( v ˆ ρ , Jv ˆ ρ ) − ds ∧ ˆ λ ( v ˆ ρ , Jv ˆ ρ ) | + | Ω( v ξ , Jv ξ ) − ˆ ω ( v ξ , Jv ξ ) | + (cid:12)(cid:12) Ω( v ξ , Jv ˆ ρ ) (cid:12)(cid:12) + (cid:12)(cid:12) Ω( v ˆ ρ , Jv ξ ) (cid:12)(cid:12) ≤ | Ω( v ˆ ρ , Jv ˆ ρ ) − ( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) | + (cid:12)(cid:12) Ω( v ξ , Jv ˆ ρ ) (cid:12)(cid:12) + (cid:12)(cid:12) Ω( v ˆ ρ , Jv ξ ) (cid:12)(cid:12) ≤ | Ω( v ˆ ρ , Jv ˆ ρ ) − ( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) | + ( ds ∧ ˆ λ + ˆ ω )( v, Jv )= (cid:12)(cid:12)(cid:12) λ ( X H ) ( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) − ( ds ∧ ˆ λ )( v ˆ ρ , Jv ˆ ρ ) (cid:12)(cid:12)(cid:12) + ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) ≤ (cid:12)(cid:12)(cid:12) ˆ λ ( X H ) − − (cid:12)(cid:12)(cid:12) ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) + ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) ≤ ( ds ∧ ˆ λ + ˆ ω )( v, Jv )In other words, we have shown that (cid:12)(cid:12) Ω( v, Jv ) − ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) (cid:12)(cid:12) ≤ ( ds ∧ ˆ λ + ˆ ω )( v, Jv )and thus ( ds ∧ ˆ λ + ˆ ω )( v, Jv ) ≤ v, Jv )for all v ∈ T ( I ǫ × M ). (cid:3) With the above estimates established, we can now turn our attention to themain result of this section. Note that the metric ˜ g occurring below is a translationinvariant metric on R × M associated to a metric on M . The norms k·k C n use thismetric. LMOST EXISTENCE FROM THE FERAL PERSPECTIVE 39
Proposition 2.9 (adiabatically degenerating constructions) . Let (cid:0) I ℓ × M, Ω , ˆ λ (cid:1) be a framed Hamiltonian energy pile, and let ˆ ρ , ξ , and ˆ ω bethe associated structures defined in Section 2, and assume, as in the conclusions ofLemma 2.5, that along { }× M the sub-bundles ˆ ρ and ξ are symplectic complementsand ˆ λ ( X H ) = 1 . Let b X = X H / ˆ λ ( X H ) as above and denote by k · k C n the C n -normon R × M with respect to the auxiliary translation invariant metric ˜ g , and the Ψ k are the embeddings as in equation (7). Let { ˇ J k } k ∈ N be a sequence of almost complexstructures on I ℓ × M which satisfy the following conditions.(D1) ˇ J k : ˆ ρ → ˆ ρ and ˇ J k : ξ → ξ for each k ∈ N ,(D2) ( ds ∧ ˆ λ + ˆ ω ) ◦ (Id × ˇ J k ) is a Riemannian metric for each k ∈ N ,(D3) there exist constants { C ′ n } n ∈ N such that sup k ∈ N k ˇ J k k C n ≤ C ′ n Then there exists an ǫ > with the following significance. For any sequence offunctions φ k : ( − ǫ, ǫ ) → (0 , which converge in C ∞ , the weakly adapted almostcomplex structures defined by φ k · J k ∂ s = b X and J k (cid:12)(cid:12) ξ := ˇ J k (cid:12)(cid:12) ξ (45) satisfy the following properties(E1) For each v ∈ T ( I ǫ × M ) we have ( ds ∧ ˆ λ + ˆ ω )( v, J k v ) ≤ v, J k v ) . (E2) There exists a sequence { K n } ∞ n =0 of positive constants so that with respect tothe auxiliary translation invariant metric ˜ g on R × M it holds that sup k ∈ N k (Ψ k ) ∗ J k k C n ≤ K n for each n ∈ N .In particular, if ǫ ′ ∈ (0 , ǫ ) and φ k (cid:12)(cid:12) ( − ǫ ′ ,ǫ ′ ) → , then the almost complex structures J k are adiabatically degenerating on I ǫ ′ × M in the sense of Definition 2.8, and on I ǫ × M these almost complex structures are all tame.Proof. Fix an auxiliary Ω-compatible almost complex structure J on I ℓ × M , andlet g be the associated background Riemannian metric g = Ω ◦ (Id × J ). Pick an ℓ ′ ∈ (0 , ℓ ) and consider the sequence ( ˇ J k ). We define C := sup k ∈ N |i ˇ J k h| ( I ℓ ′ × M, Ω , ˆ λ, ˆ ω,J ) where |i J h| is defined as in equation (34). We note that C is finite because ofhypothesis (D3). We then apply Proposition A.2 which guarantees the existence ofan ǫ > J k for which J k (cid:12)(cid:12) ξ = ˇ J k (cid:12)(cid:12) ξ .To establish (E2), assume that the φ k have been fixed so they converge in C ∞ .Define the J k as in equation (45). To estimate the norms of the (Ψ k ) ∗ J k we firstnote that it is sufficient to establish C n bounds for((Ψ k ) ∗ J k ) ∂ ˇ s and ((Ψ k ) ∗ J k ) (cid:12)(cid:12) ξ , however each of these bounds are immediately obtained as a consequence of the factthat the φ k are converging in C ∞ together with hypothesis (D3). The remainingdesired conclusions are immediate. This completes the proof of Proposition 2.9. (cid:3) A.2.
Extra Definitions.
For the convenience of the reader, we provide a fewdefinitions here which are used indirectly. The reader should consult [8], whichcontains further discussions of this concept.
Definition A.3 (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 . Definition A.4 (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 ( b J, ˆ g ) on I × M is adapted to this realized Hamiltonian homotopyprovided the following hold.(1) b J∂ t = b X .(2) b J : ˆ ξ → ˆ ξ .(3) ˆ g = ( dt ∧ ˆ λ + ˆ ω )( · , b J · ) .(4) if I is unbounded, then there exists a neighborhood of {±∞} × M on whichthe restriction b J (cid:12)(cid:12) ˆ ξ is invariant under the flow of ∂ t . References [1] C. Abbas and H. Hofer,
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E-mail address : [email protected] Helmut Hofer, School of Mathematics, Institute for Advanced Study
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