aa r X i v : . [ m a t h . S G ] J un Notes on J -Holomorphic Maps Aleksey Zinger ∗ June 2, 2017
Abstract
These notes present a systematic treatment of local properties of J -holomorphic maps andof Gromov’s convergence for sequences of such maps, specifying the assumptions needed forall statements. In particular, only one auxiliary statement depends on the manifold beingsymplectic. The content of these notes roughly corresponds to Chapters 2 and 4 of McDuff-Salamon’s book on the subject. Contents J -holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 The Monotonicity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 J -holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Global structure of J -holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Energy bound on long cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 J -holomorphic maps 37 ∗ Partially supported by NSF grants 0846978 and 1500875 Introduction
Gromov’s introduction [6] of pseudoholomorphic curves techniques into symplectic topology hasrevolutionized this field and led to its numerous connections with algebraic geometry. The ideasput forward in [6] have been further elucidated and developed in [14, 17, 11, 15, 16, 10] and inmany other works. The most comprehensive introduction to the subject of pseudoholomorphiccurves is without a doubt the monumental book [12]. Chapters 2 and 4 of this book concern two ofthe three fundamental building blocks of this subject, the local structure of J -holomorphic mapsand Gromov’s convergence for sequences of J -holomorphic maps. The present notes contain analternative systematic exposition of these two topics with generally sharper specification of theassumptions needed for each statement. Chapter 3 and Sections 6.2 and 6.3 in [12] concern thethird fundamental building block of the subject, transversality for J -holomorphic maps. A morestreamlined and general treatment of this topic is the concern of [19].The present notes build on the lecture notes on J -holomorphic maps written for the class the authortaught at Stony Brook University in Spring 2014. The lectures themselves were based on the hand-written notes he made while studying [11] back in graduate school and were also influenced by themore thorough exposition of the same topics in [12]. The author would like to thank D. McDuffand D. Salamon for the time and care taken in preparing and updating these books, the studentsin the Spring 2014 class for their participation that guided the preparation of the original versionof the present notes, and X. Chen for thoughtful comments during the revision process. A (smooth)
Riemann surface (without boundary) is a pair (Σ , j ) consisting of a smooth two-dimensional manifold Σ (without boundary) and a complex structure j in the fibers of T Σ. A nodalRiemann surface is a pair (Σ , j ) obtained from a Riemann surface ( e Σ , j ) by identifying pairs of dis-tinct points of e Σ in a discrete subset S Σ (with no point identified with more than one other point);see the left-hand sides of Figures 1 and 2. The pair ( e Σ , j ) is called the normalization of (Σ , j ); theimages of the points of S Σ in Σ are called the nodes of Σ. We denote their complement in Σ by Σ ∗ .An irreducible component of (Σ , j ) is the image of a topological component of e Σ in Σ. Let a (Σ) = 2 − χ ( e Σ) + | S Σ | , where χ ( e Σ) is the Euler characteristic of e Σ, be the ( arithmetic ) genus of Σ. An equivalence betweenRiemann surfaces (Σ , j ) and (Σ ′ , j ′ ) is a homeomorphism h : Σ −→ Σ ′ induced by a biholomor-phic map e h from ( e Σ , j ) to ( e Σ ′ , j ′ ). We denote by Aut(Σ , j ) the group of automorphisms, i.e. self-equivalences, of a Riemann surface (Σ , j ).Let ( X, J ) be an almost complex manifold. If (Σ , j ) is a Riemann surface, a smooth map u : Σ −→ X is called J -holomorphic map if d u ◦ j = J ◦ d u : T Σ −→ u ∗ T X. A J -holomorphic map from a nodal Riemann surface (Σ , j ) is a tuple (Σ , j , u ), where u : Σ −→ X is acontinuous map induced by a J -holomorphic map e u : e Σ −→ X ; see Figures 1 and 2. An equivalence between J -holomorphic maps (Σ , j , u ) and (Σ ′ , j ′ , u ′ ) is an equivalence h : (Σ , j ) −→ (Σ ′ , j ′ )2 (Σ , j ) = (Σ , j ) ∨ (Σ , j ) (Σ , j ) = (Σ , j ) z ∗ z ∗ Figure 1: A stable J -holomorphic mapbetween the underlying Riemann surfaces such that u = u ′ ◦ h . We denote by Aut(Σ , j , u ) the groupof automorphisms, i.e. self-equivalences, of a J -holomorphic map (Σ , j , u ). A J -holomorphic map(Σ , j , u ) is called stable if (Σ , j ) is compact and Aut(Σ , j , u ) is a finite group.The Riemann surface (Σ , j ) on the left-hand side of Figure 1 is obtained by identifying the markedpoints of two copies of a smooth elliptic curve (Σ , j , z ∗ ), i.e. a torus with a complex structureand a marked point. The Riemann surface (Σ , j ) with the marked point z ∗ is biholomorphic to C / Λ with the marked point 0 for some lattice Λ ⊂ C and thus has an automorphism of order 2 thatpreserves z ∗ (it is induced by the map z −→ − z on C ). This is the only non-trivial automorphism of(Σ , j ) preserving z ∗ if j is generic; in special cases, the group of such automorphisms is either Z or Z . Each automorphism of (Σ , j ) preserving z ∗ gives rise to an automorphism of (Σ , j ) fixingone of the irreducible components. There is also an automorphism of (Σ , j ) which interchanges thetwo irreducible components of Σ. Since it does not commute with the automorphisms preservingone of the components, Aut(Σ , j ) ≈ D in most cases and contains D in the special cases. If u : Σ −→ Σ is the identity on each irreducible component, (Σ , j , u ) is a stable J -holomorphic map;the interchange of the two irreducible components is then the only non-trivial automorphism of(Σ , j , u ). The J -holomorphic maps u : Σ −→ Σ obtained by sending either or both irreduciblecomponents of Σ to z ∗ instead are also stable, but have different automorphism groups. If (Σ , j )were taken to be the Riemann sphere P , the J -holomorphic map u : Σ −→ Σ restricting to theidentity on each copy of Σ would still be stable. However, a map u : Σ −→ Σ sending either copyof Σ to z ∗ would not be stable, since the group of automorphisms of P fixing a point is a complextwo-dimensional submanifold of PSL .Let (Σ , j ) be a compact connected Riemann surface of genus g . If g ≥
2, then Aut(Σ , j ) is a finitegroup. If g = 1, then Aut(Σ , j ) is an infinite group, but its subgroup fixing any point is finite. If g = 0, then the subgroup of Aut(Σ , j ) fixing any pair of points is infinite, but the subgroup fixing anytriple of points is trivial. If in addition ( X, J ) is an almost complex manifold and u : Σ −→ X is anon-constant J -holomorphic map, then the subgroup of Aut(Σ , j ) consisting of the automorphismssuch that u = u ◦ h is finite; this is an immediate consequence of Corollary 3.4. If (Σ , j ) is a compactnodal Riemann surface, a J -holomorphic map (Σ , j , u ) is thus stable if and only if • every genus 1 topological component of the normalization e Σ of Σ such that u restricts to aconstant map on its image in Σ contains at least 1 element of S Σ and • every genus 0 topological component of e Σ such that u restricts to a constant map on its imagein Σ contains at least 3 elements of S Σ . 3 .2 Gromov’s topology Given a Riemann surface (Σ , j ), a Riemannian metric g on a smooth manifold X determines the energy E g ( f ) for every smooth map f : Σ −→ X ; see (2.5) and (2.6). The fundamental insight in [6]that laid the foundations for the pseudoholomorphic curves techniques in symplectic topologyand for the moduli spaces of stable maps and related curve-parametrizing objects in algebraicgeometry is that a sequence of stable J -holomorphic maps (Σ i , j i , u i ) into a compact almost complexmanifold ( X, J ) with lim inf i −→∞ (cid:16)(cid:12)(cid:12) π (Σ i ) (cid:12)(cid:12) + a (Σ i )+ E g ( u i ) (cid:17) < ∞ (1.1)has a subsequence converging in a suitable sense to another stable J -holomorphic map.The notion of Gromov’s convergence of a sequence of stable J -holomorphic maps (Σ i , j i , u i ) toanother stable J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) comes down to(GC1) | π (Σ i ) | = | π (Σ ∞ ) | and a (Σ i ) = a (Σ ∞ ) for all i large,(GC2) (Σ ∞ , j ∞ ) is at least as singular as (Σ i , j i ) for all i large,(GC3) the energy is preserved, i.e. E g ( u i ) −→ E g ( u ∞ ) as i −→ ∞ , and(GC4) u i converges to u ∞ uniformly in the C ∞ -topology on compact subsets of Σ ∗∞ .Most applications of the pseudoholomorphic curves techniques in symplectic topology involve J -holomorphic maps from the Riemann sphere P . This is a special case of the situation whenthe complex structures j i on the domains Σ i of u i are fixed. The condition (GC4) can then beformally stated in a way clearly indicative of the rescaling procedure of [6]. Definition 1.1 (Gromov’s Compactness I) . Let (
X, J ) be an almost complex manifold withRiemannian metric g and (Σ , j ) be a compact Riemann surface. A sequence (Σ , j , u i ) of stable J -holomorphic maps converges to a stable J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) if(1) (Σ ∞ , j ∞ ) is obtained from (Σ , j ) by identifying a point on each of ℓ trees of Riemann spheres P ,for some ℓ ∈ Z ≥ , with distinct points z ∗ , . . . , z ∗ ℓ ∈ Σ,(2) E g ( u ∞ ) = lim i −→∞ E g ( u i ),(3) there exist h i ∈ Aut(Σ , j ) with i ∈ Z + such that u i ◦ h i converges to u ∞ uniformly in the C ∞ -topology on compact subsets of Σ −{ z ∗ , . . . , z ∗ ℓ } ,(4) for each z ∗ , . . . , z ∗ ℓ ∈ Σ ⊂ Σ ∞ and all i ∈ Z + sufficiently large, there exist a neighborhood U j ⊂ Σof z ∗ j , an open subset U j ; i ⊂ C , and a biholomorphic map ψ j ; i : U j ; i −→ U j such that(4a) U i ⊂ U i +1 and C = S ∞ i =1 U j ; i for every j = 1 , . . . , ℓ ,(4b) u i ◦ h i ◦ ψ j ; i converges to u ∞ uniformly in the C ∞ -topology on compact subsets of thecomplement of the nodes ∞ , w ∗ j ;1 , . . . , w ∗ j ; k j in the sphere P j attached at z ∗ j ∈ Σ,(4c) condition (4) applies with Σ, ( z ∗ , . . . , z ∗ ℓ ), and u i ◦ h i replaced by P , ( w ∗ j ;1 , . . . , w ∗ j ; k j ), and u i ◦ h i ◦ ψ j ; i , respectively, for each j = 1 , . . . , ℓ .4 ∞ = Xu ∞ Σ z ∗ ∞ z ∗ ∞ w ∗ w ∗ ∞∞ Figure 2: Gromov’s limit of a sequence of J -holomorphic maps u i : Σ −→ X An example of a possible limiting map with ℓ = 2 trees of spheres is shown in Figure 2. The recursivecondition (4) in Definition 1.1 is equivalent to the Rescaling axiom in [12, Definition 5.2.1] onsequences of automorphisms φ iα of P ; they correspond to compositions of the maps ψ j ; i associatedwith different irreducible components of Σ ∞ . The single energy condition (2) in Definition 1.1is replaced in [12, Definition 5.2.1] by multiple conditions of the Energy axiom. These multipleconditions are equivalent to (2) if the other three axioms in [12, Definition 5.2.1] are satisfied.
Theorem 1.2 (Gromov’s Compactness I) . Let ( X, J ) be a compact almost complex manifold withRiemannian metric g , (Σ , j ) be a compact Riemann surface, and u i : Σ −→ X be a sequence ofnon-constant J -holomorphic maps. If lim inf E g ( u i ) < ∞ , then the sequence (Σ , j , u i ) contains asubsequence converging to some stable J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) in the sense of Defini-tion 1.1. This theorem is established in Section 5.3 by assembling together a number of geometric statementsobtained earlier in these notes. In Section 5.4, we relate the convergence notion of Definition 1.1 inthe case of holomorphic maps from CP to CP n , which can always be represented by ( n +1)-tuplesof homogeneous polynomials in two variables, to the behavior of the linear factors of the associatedpolynomials.The convergence notion of Definition 1.1 can be equivalently reformulated in terms of deformationsof the limiting domain (Σ ∞ , j ∞ ) so that it readily extends to sequences of stable J -holomorphicmaps with varying complex structures j i on the domains Σ i . This was formally done in the algebraicgeometry category by [4], several years after this perspective had been introduced into the fieldinformally, and adapted to the almost complex category by [10]. We summarize this perspectivebelow.Let (Σ , j ) be a nodal Riemann surface. A flat family of deformations of (Σ , j ) is a holomorphic map π : U −→ ∆, where U is a complex manifold and ∆ ⊂ C N is a neighborhood of 0, such that • π − ( λ ) is a nodal Riemann surface for each λ ∈ C n and π − (0) = (Σ , j ), • π is a submersion outside of the nodes of the fibers of π , • for every λ ∗ ≡ ( λ ∗ , . . . , λ ∗ N ) ∈ ∆ and every node z ∗ ∈ π − ( λ ∗ ), there exist i ∈ { , . . . , N } with λ i = 0, neighborhoods ∆ λ ∗ of λ ∗ in ∆ and U z ∗ of z ∗ in U , and a holomorphic mapΨ : U z ∗ −→ (cid:8)(cid:0) ( λ , . . . , λ N ) , x, y (cid:1) ∈ ∆ λ ∗ × C : xy = λ i (cid:9) such that Ψ is a homeomorphism onto a neighborhood of ( λ ∗ , ,
0) and the composition of Ψwith the projection to ∆ λ ∗ equals π | U z ∗ . 5 λ λ ∆ U π B δ ( z ∗ ) B δ ( z ∗ ) z ∗ z ∗ Σ ∞ Σ λ Σ λ Figure 3: A complex-geometric presentation of a flat family of deformations of (Σ ∞ , j ∞ ) = π − (0)and a differential-geometric presentation of the domains of the maps u i in Definition 1.3.If π : U −→ ∆ is a flat family of deformations of (Σ , j ) and Σ is compact, there exists a neighborhood U ∗ ⊂ U of Σ ∗ ⊂ π − (0) such that π | U ∗ : U ∗ −→ ∆ ≡ π ( U ∗ ) ⊂ ∆is a trivializable Σ ∗ -fiber bundle in the smooth category. For each λ ∈ ∆ , let ψ λ : Σ ∗ −→ π − ( λ ) ∩U ∗ be the corresponding smooth identification. If λ i ∈ ∆ is a sequence converging to 0 ∈ ∆ and u i : π − ( λ i ) −→ X is a sequence of continuous maps that are smooth on the complements of thenodes of π − ( λ i ), we say that the sequence u i converges to a smooth map u : Σ ∗ −→ X u .c.s. if thesequence of maps u i ◦ ψ λ i : Σ ∗ −→ X converges uniformly in the C ∞ -topology on compact subsets of Σ ∗ . This notion is independent ofthe choices of U ∗ and trivialization of π | U ∗ . Definition 1.3 (Gromov’s Convergence II) . Let (
X, J ) be an almost complex manifold with Rie-mannian metric g . A sequence (Σ i , j i , u i ) of stable J -holomorphic maps converges to a stable J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) if E g ( u ∞ ) −→ E g ( u i ) as i −→ ∞ and there exist(a) a flat family of deformations π : U −→ ∆ of (Σ ∞ , j ∞ ),(b) a sequence λ i ∈ ∆ converging to 0 ∈ ∆, and(c) equivalences h i : π − ( λ i ) −→ (Σ i , j i )such that u i ◦ h i converges to u ∞ | Σ ∗∞ u.c.s.By the compactness of Σ ∞ , the notion of convergence of Definition 1.3 is independent of the choiceof metric g on X . It is illustrated in Figure 3. If the Riemann surfaces (Σ i , j i ) are smooth, thelimiting Riemann surface (Σ ∞ , j ∞ ) is obtained by pinching some disjoint embedded circles in thesmooth two-dimensional manifold Σ underlying these Riemann surfaces.6f (Σ i , j i ) = (Σ , j ) for all i as in Definition 1.1, only contractible circles are pinched to produce Σ ∞ ;it then consists of Σ with trees of spheres attached. The family π : U −→ ∆ is obtained by startingwith the family π : U ≡ C × Σ −→ C , then blowing up U at a point of { } × Σ to obtain a family π : U −→ C with the central fiberΣ ≡ π − (0) consisting of Σ with P attached, then blowing up a smooth point of Σ , and so on.The number of blowups involved is precisely the number of nodes of Σ ∞ , i.e. four in the case ofFigure 2 and two in the case of Figure 3. The pinched annuli on the right-hand side of Figure 3correspond to φ α ( B δ ( z αβ )) ∪ φ β ( B δ ( z βα )) in the notation of [12, Chapters 4,5].With the setup of Definition 1.3, let B δ ( z ∗ ) ⊂ U denote the ball of radius δ ∈ R + around a point z ∗ ∈ U with respect to some metric on U . Then,lim δ −→ lim i −→∞ diam g (cid:0) u i (cid:0) h i ( π − ( λ i ) ∩ B δ ( z ∗ )) (cid:1)(cid:1) = 0 ∀ z ∗ ∈ Σ ∞ . (1.2)This is immediate from the last condition in Definition 1.3 if z ∗ ∈ Σ ∗∞ . If z ∗ ∈ Σ ∞ − Σ ∗∞ is a nodeof Σ ∞ , (1.2) is a consequence of both convergence conditions of Definition 1.3 and the maps u i being J -holomorphic. It is a reflection of the fact that bubbling or any other kind of erratic C -behavior of a sequence of J -holomorphic maps requires a nonzero amount of energy in the limit,but the two convergence conditions of Definition 1.3 ensure that all limiting energy is absorbedby u | Σ ∗∞ and thus none is left for bubbling around the nodes of Σ ∞ . An immediate implicationof (1.2) is that u i ( h i ( π − ( λ i ) ∩ B δ ( z ∗ ))) is contained in a geodesic ball around u ∞ ( z ∗ ) in X . Thus, u i ∗ (cid:2) Σ i (cid:3) = u ∞∗ [Σ ∞ ] ∈ H ( X ; Z )for all i ∈ Z + sufficiently large. If Σ ∞ is a tree of spheres (and thus so is each Σ i ), then u i with i sufficiently large lies in the equivalence class in π ( X ) determined by u ∞ for the same reason. Theorem 1.4 (Gromov’s Compactness II) . Let ( X, J ) be a compact almost complex manifold withRiemannian metric g and (Σ i , j i , u i ) be a sequence of stable J -holomorphic maps into a compactalmost complex manifold ( X, J ) . If it satisfies (1.1), then it contains a subsequence converging tosome stable J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) in the sense of Definition 1.3. This theorem is obtained by combining the compactness of the Deligne-Mumford moduli spaces M , of stable (possibly) nodal elliptic curves and M g of stable nodal genus g ≥ i , j i ) is a smooth connected Riemann surface of genus g ≥ g = 0 case is treated byTheorem 1.2). If g = 1, we add a marked point to each domain (Σ i , j i ) and take a subsequenceconverging in M , to the equivalence class of some stable nodal elliptic curve (Σ ′∞ , j ′∞ , z ′∞ ). If g ≥ i , j i ) converging in M g to the equivalence class of some stable nodalgenus g curve (Σ ′∞ , j ′∞ ). This ensures the existence of a flat family of deformations π ′ : U ′ −→ ∆ ′ of(Σ ′∞ , j ′∞ ), of a sequence λ ′ i ∈ ∆ ′ converging to 0 ∈ ∆ ′ , and of equivalences h i : π ′− ( λ ′ i ) −→ (Σ i , j i ).The associated neighborhood U ′∗ of Σ ′∗∞ in U ′ can be chosen so that π ′− ( λ ) −U ′∗ consists of finitelymany circles for every λ ′ ∈ ∆ ′ sufficiently small. The complement of the image of the associatedidentifications ψ ′ λ : Σ ′∗∞ −→ π ′− ( λ ) ∩U ′∗ in π ′− ( λ ) has the same property. 7ne then applies the construction in the proof of Theorem 1.2 to the sequence of J -holomorphicmaps u i ◦ h ′ i : Σ ′∗∞ −→ X to obtain a J -holomorphic map e u ′∞ from the normalization e Σ ′∞ of Σ ∞ and finitely J -holomorphicmaps from trees of P . Each of these trees will have one or two special points that are asso-ciated with points of e Σ ′∞ (the latter happens if bubbling occurs at a preimage of a node of Σ ′∞ in e Σ ′∞ ). Identifying these trees with the corresponding points of e Σ ′∞ as in the proof of Theorem 1.2,we obtain a J -holomorphic map (Σ ∞ , j ∞ , u ∞ ) satisfying the requirements of Definition 1.3. It isnecessarily stable if g ≥
2, or Σ ′∞ is smooth, or Σ ∞ contains a separating node. Otherwise, theidentifications h ′ i may first need to be reparametrized to ensure that either the limiting map e u ′∞ isnot constant or the sequence u i ◦ h i produces a bubble at least one smooth point of e Σ ′∞ .A k -marked Riemann surface is a tuple (Σ , j , z , . . . , z k ) such that (Σ , j ) is a Riemann surfaceand z , . . . , z k ∈ Σ ∗ are distinct points. If ( X, J ) is an almost complex manifold, a k -marked J -holomorphic map into X is a tuple (Σ , j , z , . . . , z k , u ), where (Σ , j , z , . . . , z k ) is k -marked Rie-mann surface and (Σ , j , u ) is a J -holomorphic map into X . The degree of such a map is thehomology class A = u ∗ [Σ] ∈ H ( X ; Z ) . The notions of equivalence, stability, and convergence as in Definition 1.3 and the above convergenceargument for smooth domains (Σ i , j i ) readily extend to k -marked J -holomorphic maps. The generalcase of Theorem 1.4, including its extension to stable marked maps, is then obtained by • passing to a subsequence of (Σ i , j i , u i ) with the same topological structure of the domain, • viewing it as a sequence of tuples of J -holomorphic maps with smooth domains with an additionalmarked point for each preimage of the nodes in the normalization, and • applying the conclusion of the above argument to each component of the tuple. The natural extension of Definition 1.3 to marked J -holomorphic maps topologizes the modulispace M g,k ( X, A ; J ) of equivalence classes of stable degree A k -marked genus g J -holomorphicmaps into X for each A ∈ H ( X ; Z ). The evaluation mapsev i : M g,k ( X, A ; J ) −→ X, (Σ , j , z , . . . , z k , u ) −→ u ( z i ) , are continuous with respect to this topology. If 2 g + k ≥
3, there is a continuous map f : M g,k ( X, A ; J ) −→ M g,k to the Deligne-Mumford moduli space of stable k -marked genus g nodal curves obtained by forget-ting the map u and then contracting the unstable components of the domain.There is a continuous map f k +1 : M g,k +1 ( X, A ; J ) −→ M g,k ( X, A ; J ) (1.3)8 Σ z z z z ′ z ′ z ′ z ′ Σ ′ Figure 4: Section s of the fibration (1.3) with k = 3obtained by forgetting the last marked point z k +1 and then contracting the components of thedomain to stabilize the resulting k -marked J -holomorphic map. For each i = 1 , . . . , k , this fibrationhas a natural continuous section s i : M g,k ( X, A ; J ) −→ M g,k +1 ( X, A ; J )described as follows. For a k -marked nodal Riemann surface (Σ , j , z , . . . , z k ), let (Σ ′ , j ′ , z , . . . , z k +1 )be the ( k +1)-marked nodal Riemann surface so that (Σ ′ , j ′ ) consists of (Σ , j ) with P attached at z i , z ′ , z ′ i ∈ P , and z ′ j = z j ∈ Σ for all j = 1 , . . . , k different from k ; see Figure 4. We define s i (cid:0) [Σ , j , z , . . . , z k , u ] (cid:3) = (cid:2) Σ ′ , j ′ , z ′ , . . . , z ′ k +1 , u ′ (cid:3) , with (Σ ′ , j ′ , z ′ , . . . , z ′ k +1 ) as described and u ′ extending u over the extra P by the constant mapwith value u ( z i ). The pullback L i −→ M g,k ( X, A ; J )of the vertical tangent line bundle of (1.3) by s i is called the universal tangent line bundle at the i -th marked point. Let ψ i = c ( L ∗ i ) be the i -th descendant class .A remarkable property of Gromov’s topology which lies behind most of its applications is thatthe moduli space M g,k ( X, A ; J ) is Hausdorff and has a particularly nice deformation-obstructiontheory. In the algebraic-geometry category, the latter is known as a perfect two-term deformation-obstruction theory . In the almost complex category, this is reflected in the existence of an atlas offinite-dimensional approximations in the terminology of [10] or of an atlas of Kuranishi charts in theterminology of [10].If ( X, J ) is an almost complex manifold and J is tamed by a symplectic form ω , then the energy E g ( u ) of degree A J -holomorphic map u with respect to the metric g determined by J and ω is ω ( A ); see (2.7). In particular, it is the same for all elements of the moduli space M g,k ( X, A ; J ).If in addition X is compact, then Theorem 1.4 implies that this moduli space is also compact.Combining this with the remarkable property of the previous paragraph, the constructions of[1, 9, 10, 3] endow M g,k ( X, A ; J ) with a virtual fundamental class . It depends only on ω , in asuitable sense, and not an almost complex structure J tamed by ω . This class in turn gives rise to Gromov-Witten invariants of (
X, ω ): (cid:10) τ a α , . . . , τ a k α k (cid:11) Xg,A ≡ (cid:10)(cid:0) ψ a ev ∗ α ) . . . (cid:0) ψ a k k ev ∗ k α k ) , (cid:2) M g,k ( X, A ; J ) (cid:3) vir (cid:11) ∈ Q for all a i ∈ Z ≥ and α i ∈ H ∗ ( X ; Q ). 9 Preliminaries
An outline of these notes with an informal description of the key statements appears in Section 2.1;Figure 5 indicates primary connections between these statements. Sections 2.2 introduces the mostfrequently used notation and terminology and makes some basic observations.
The main technical statement of Section 3 of these notes and of Chapter 2 in [12] is the
CarlemanSimilarity Principle ; see Proposition 3.1. It yields a number of geometric conclusions about the localbehavior of a J -holomorphic map u : Σ −→ X from a Riemann surface (Σ , j ) into an almost complexmanifold ( X, J ). For example, for every z ∈ Σ contained in a component of Σ on which u is notconstant, the ℓ -th derivative of u at z in a chart around u ( z ) does not vanish for some ℓ ∈ Z + ; seeCorollary 3.3. We denote by ord z u ∈ Z + the minimum of such integers ℓ and call it the order of u at z ; it is independent of the choice of a chart around u ( z ). If u is constant on the component of Σon containing z , we set ord z u = 0. A point z ∈ u is singular , i.e. d z u = 0, if and only if ord z u = 1.If u is not constant on every connected component of Σ, the singular points of u and the preimagesof a point x ∈ X are discrete subsets of Σ; see Corollary 3.4. In the case Σ is compact, the secondstatement of Corollary 3.4 implies thatord x u ≡ X z ∈ u − ( x ) ord z u ∈ Z ≥ ∀ x ∈ X ; (2.1)we call this number the order of u at x . If x Im( u ), then ord x u = 0. By Corollary 3.11, thenumber (2.1) is seen by the behavior of the energy (2.5) of u and its restrictions to open subsetsof Σ. This observation underpins the Monotonicity Lemma for J -holomorphic maps, which boundsbelow the energy required to “escape” from a small ball in X ; see Proposition 3.12.The main technical statement of Section 4 of these notes and of Chapter 4 in [12] is the Mean ValueInequality . It bounds the pointwise differentials d z u of a J -holomorphic map u from (Σ , j ) into ( X, J )of sufficiently small energy E g ( u ) by E g ( u ), i.e. by the L -norm of d u , from above and immediatelyyields a bound on the energy of non-constant J -holomorphic maps from S into ( X, J ) from below;see Proposition 4.1 and Corollary 4.2, respectively. The Mean Value Inequality also implies thatthe energy of a J -holomorphic map u from a cylinder [ − R, R ] × S carried by [ − R + T, R − T ] × S and the diameter of the image of this middle segment decay at least exponentially with T , providedthe overall energy of u is sufficiently small. As shown in the proof of Proposition 5.5, this techni-cal implication ensures that the energy is preserved under Gromov’s convergence and the resulting bubbles connect .Another important implication of Proposition 4.1 is that a continuous map from a Riemann surface(Σ , j ) into an almost complex manifold ( X, J ) which is holomorphic outside of a discrete collectionof points and has bounded energy is in fact holomorphic on all of Σ; see Proposition 4.8. Thisconclusion plays a central role in the proof of Lemma 5.4. Theorem 1.2 is deduced from Lemma 5.4and Proposition 5.5 in Section 5.3.Combined with Proposition 3.1 and some of its corollaries, Proposition 4.1 implies that every non-constant J -holomorphic map from a compact Riemann surface (Σ , j ) factors through a somewhere (cid:15) (cid:15) " " ❋❋❋❋❋❋❋❋ z z ttttttttt (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ $ $ ■■■■■■■■■ (cid:15) (cid:15) / / o o | | ①①①①①①①① z z ✉✉✉✉✉✉✉✉✉ (cid:4) (cid:4) ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ / / / / (cid:15) (cid:15) ■■ $ $ ■■■■■■ / / J -holomorphic maps 4.8 Regularity of J -holomorphic maps4.17 Bounds on long cylinders 5.1 Removal of Singularity5.2-5.4 Bubbling 5.5 Gromov’s convergenceFigure 5: Connections between the main statements leading to Theorem 1.2 injective J -holomorphic map from a compact Riemann surface (Σ ′ , j ′ ); see Proposition 4.11. Theproof of this statement with X compact appears in Chapter 2 of [12], but uses the RemovalSingularities Theorem proved in Chapter 4 of [12]. Let (Σ , j ) be a Riemann surface, V be a vector bundle over Σ, and µ, η ∈ Γ (cid:0) Σ; T ∗ Σ ⊗ R V (cid:1) and g ∈ Γ (cid:0) Σ; T ∗ Σ ⊗ ⊗ R V (cid:1) . For a local coordinate z = s + i t , define g ( µ ⊗ j η ) = (cid:0) g (cid:0) µ ( ∂ s ) , η ( ∂ s ) (cid:1) + g (cid:0) µ ( ∂ t ) , η ( ∂ t ) (cid:1)(cid:1) d s ∧ d t ,g ( µ ∧ j η ) = (cid:0) g (cid:0) µ ( ∂ s ) , η ( ∂ t ) (cid:1) − g (cid:0) µ ( ∂ t ) , η ( ∂ s ) (cid:1)(cid:1) d s ∧ d t . (2.2)By a direct computation, the 2-forms g ( µ ⊗ j η ) and g ( µ ∧ j η ) are independent of the choice of localcoordinate z = s + i t . Thus, (2.2) determines global 2-forms on Σ (which depend on the choice of j ).We denote by i the standard complex structure on C and by J C n the standard complex structureson C n and T C n . For an almost complex structure J and a 2-form ω on a manifold X , we define a2-tensor and a 2-form on X by g J ( v, v ′ ) = 12 (cid:0) ω ( v, J v ′ ) − ω ( J v, v ′ ) (cid:1) ,ω J ( v, v ′ ) = 12 (cid:0) ω ( J v, J v ′ ) − ω ( v, v ′ ) (cid:1) ∀ v, v ′ ∈ T x X, x ∈ X. (2.3)We note that g J ( v, v ) + g J ( v ′ , v ′ ) = 2 ω ( v, v ′ ) + g J ( v + J v ′ , v + J v ′ ) + 2 ω J ( v, v ′ ) ∀ v, v ′ ∈ T x X, x ∈ X. (2.4)11he 2-form ω tames J if g ( v, v ) > v ∈ T X nonzero; in such a case, ω is nondegenerate and g J is a metric. The almost complex structure J is ω -compatible if ω tames J and ω J = 0.Let X be a manifold, (Σ , j ) be a Riemann surface, and f : Σ −→ X be a smooth map. We denotethe pullbacks of a 2-tensor g and a 2-form ω on X to the vector bundle f ∗ T X over Σ also by g and ω . If g is a Riemannian metric on X and U ⊂ Σ is an open subset, let E g ( f ) ≡ Z Σ g (d f ⊗ j d f ) ∈ [0 , ∞ ] and E g ( f ; U ) ≡ E g (cid:0) f | U ) (2.5)be the energy of f and of its restriction to U . By the first equation in (2.2), E g ( f ) = 12 Z Σ | d f | g Σ ,g (2.6)is the square of the L -norm of d f with respect to the metric g on X and a metric g Σ compatiblewith j . In particular, the right-hand side of (2.6) depends on the metric g on X and on the complexstructure j on Σ, but not the metric g Σ on Σ compatible with j .Let J be an almost complex structure on a manifold X and (Σ , j ) be a Riemann surface. For asmooth map f : Σ −→ X , define¯ ∂ J f = 12 (cid:0) d f + J ◦ d f ◦ j (cid:1) ∈ Γ (cid:0) Σ; ( T ∗ Σ , j ) , ⊗ C f ∗ ( T X, J ) (cid:1) . If ω is a 2-form on X taming J and u : Σ −→ X is J -holomorphic, then E g J ( f ) = Z Σ (cid:0) f ∗ ω +2 g J ( ¯ ∂ J f ⊗ j ¯ ∂ J f )+ f ∗ ω J (cid:1) (2.7)by (2.5) and (2.4). If J is ω -compatible, the last term above vanishes. A smooth map u : Σ −→ X is J -holomorphic if ¯ ∂ J u = 0. For such a map, the last two terms in (2.7) vanish.For each R ∈ R + , denote by B R ⊂ C the open ball of radius R around the origin and let B ∗ R = B R −{ } . If in addition (
X, g ) is a Riemannian manifold and x ∈ X , let B gδ ( x ) ⊂ X be the ball of radius δ around x in X with respect to the metric g .Let ( X, J ) be an almost complex manifold and (Σ , j ) be a Riemann surface. A smooth map u : Σ −→ X is called • somewhere injective if there exists z ∈ Σ such that u − ( u ( z )) = { z } and d z u = 0, • multiply covered if u = u ′ ◦ h for some smooth connected orientable surface Σ ′ , branched cover h : Σ −→ Σ ′ of degree different from ±
1, and a smooth map u ′ : Σ ′ −→ X , • simple if it is not multiply covered.By Proposition 4.11, every J -holomorphic map from a compact Riemann surface is simple if andonly if it is somewhere injective (the if implication is trivial).12 Local Properties
We begin by studying local properties of J -holomorphic maps u from Riemann surfaces (Σ , j ) intoalmost complex manifolds ( X, J ) that resemble standard properties of holomorphic maps. None ofthe statements in Section 3 depending on X being compact; very few depend on Σ being compact. Carleman Similarity Principle , i.e. Proposition 3.1 below, is a local description of solutions of a non-linear differential equation which generalizes the equation ¯ ∂ J u = 0. It states that such solutionslook similar to holomorphic maps and implies that they exhibit many local properties one wouldexpect of holomorphic maps. Proposition 3.1 (Carleman Similarity Principle, [2, Theorem 2.2]) . Suppose n ∈ Z + , p, ǫ ∈ R + with p > , J ∈ L p ( B ǫ ; End R C n ) , C ∈ L p ( B ǫ ; End R C n ) , and u ∈ L p ( B ǫ ; C n ) are such that u (0) = 0 , J ( z ) = − Id C n , u s ( z ) + J ( z ) u t ( z ) + C ( z ) u ( z ) = 0 ∀ z = s + i t ∈ B ǫ . (3.1) Then, there exist δ ∈ (0 , ǫ ) , Φ ∈ L p ( B δ ; GL n R ) , and a J C n -holomorphic map σ : B δ −→ C n such that σ (0) = 0 , J ( z )Φ( z ) = Φ( z ) J C n , u ( z ) = Φ( z ) σ ( z ) ∀ z ∈ B δ . (3.2)By the Sobolev Embedding Theorem [18, Corollary 4.3], the assumption p > u is acontinuous function. In particular, all equations in (3.1) and in (3.2) make sense. This assumptionalso implies that the left-hand sides of the third equation in (3.1) and of the second equation in (3.2)and the right-hand side of the third equations in (3.2) lie in L p . Example 3.2.
Let c : C −→ C denote the usual conjugation. Define b J ( z , z ) = (cid:18) i − i s c i (cid:19) = (cid:18) s c (cid:19) J C (cid:18) s c (cid:19) − : C −→ C ∀ z i = s i + i t i ,u : C −→ C , u ( s + i t ) = (cid:0) z, s ) . Thus, b J is an almost complex structure on C and u is a b J -holomorphic map, i.e. it satisfies thelast condition in (3.1) with J ( z ) = b J ( u ( z )) and C ( z ) = 0. The functions σ : C −→ C , σ ( z ) = ( z, , Φ : C −→ GL R , Φ( s + i t ) = (cid:18) s c + i stz (cid:19) , satisfy (3.2). Corollary 3.3.
Let n , p , ǫ , J , C , and u be as in Proposition 3.1. If in addition J = J C n and u does not vanish to infinite order 0, then there exist ℓ ∈ Z + and α ∈ C n − such that lim z −→ u ( z ) − αz ℓ z ℓ = 0 . Proof.
This follows from (3.2) and from the existence of such ℓ and α for σ .13 orollary 3.4. Suppose ( X, J ) is an almost complex manifold, (Σ , j ) is a Riemann surface, and u : Σ −→ X is a J -holomorphic map. If u is not constant on every connected component of Σ , thenthe subset u − (cid:0) { u ( z ) : z ∈ Σ , d z u = 0 } (cid:1) ⊂ Σ is discrete. If in addition x ∈ X , the subset u − ( x ) ⊂ Σ is also discrete.Proof. The first and third equations in (3.2) immediately imply the second claim (but not the first,since Φ may not be in C ). The first claim follows from Corollary 3.3 and Taylor’s formula for u (as well as from Corollary 3.6).Before establishing the full statement of Proposition 3.1, we consider a special case. Lemma 3.5.
Suppose n ∈ Z + and p, ǫ ∈ R + are as in Proposition 3.1, A ∈ L p ( B ǫ ; End C C n ) , and u ∈ L p ( B ǫ ; C n ) are such that u (0) = 0 , u s + J C n u t ( z ) + A ( z ) u ( z ) = 0 ∀ z = s + i t ∈ B ǫ . (3.3) Then, there exist δ ∈ (0 , ǫ ) , Φ ∈ L p ( B δ ; GL n C ) , a J C n -holomorphic map σ : B δ −→ C n such that σ (0) = 0 , Φ(0) = Id C n , u ( z ) = Φ( z ) σ ( z ) ∀ z ∈ B δ . (3.4) Proof.
For each δ ∈ [0 , ǫ ], we define A δ ∈ L p ( S ; End C C n ) by A δ ( z ) = ( A ( z ) , if z ∈ B δ ;0 , otherwise; D δ : L p ( S ; End C C n ) −→ L p ( S ; ( T ∗ S ) , ⊗ C End C C n ) by D δ Θ = (cid:0) Θ s + J C n Θ t + A δ Θ (cid:1) d¯ z . Since the cokernel of D = 2 ¯ ∂ is isomorphic H ( S ; C ) ⊗ C End C C n , D is surjective and the homo-morphism e D : L p ( S ; End C C n ) −→ L p ( S ; ( T ∗ S ) , ⊗ C End C C n ) ⊕ End C C n , Θ −→ (cid:0) D Θ , Θ(0) (cid:1) , is an isomorphism. Since (cid:13)(cid:13) D δ Θ − D Θ (cid:13)(cid:13) L p ≤ k A δ k L p k Θ k C ≤ C k A δ k L p k Θ k L p ∀ Θ ∈ L p ( S ; End C C n )and k A δ k L p −→ δ −→
0, the homomorphism e D δ : L p ( S ; End C C n ) −→ L p ( S ; ( T ∗ S ) , ⊗ C End C C n ) ⊕ End C C n , Θ −→ (cid:0) D δ Θ , Θ(0) (cid:1) , is also an isomorphism for δ > δ = D − δ (0 , Id C n ). Since D δ is an isomorphism, (cid:13)(cid:13) Θ δ − Id C n (cid:13)(cid:13) C ≤ C (cid:13)(cid:13) Θ δ − Id C n (cid:13)(cid:13) L p ≤ C ′ (cid:13)(cid:13) D δ (Θ δ − Id C n ) (cid:13)(cid:13) L p = C ′ (cid:13)(cid:13) A δ (cid:13)(cid:13) L p . Since k A δ k L p −→ δ −→
0, Θ δ ∈ L p ( B δ ; GL n C ). By (3.3) and D δ Θ δ = 0, the function σ ≡ Θ − δ u satisfies σ (0) = 0 , σ s + J C n σ t = 0 ∀ z ∈ B δ , i.e. σ is J C n -holomorphic, as required. 14 roof of Proposition 3.1 . (1) Since B ǫ is contractible, the complex vector bundles u ∗ ( T C n , J C n )and u ∗ ( T C n , J ) over B ǫ are isomorphic. Thus, there existsΨ ∈ L p ( B ǫ ; GL n R ) s.t. J ( z )Ψ( z ) = Ψ( z ) J C n ∀ z ∈ B ǫ . Let v = Ψ − u . By the assumptions on u , v ∈ L p ( B ǫ ; C n ) and v (0) = 0 , v s ( z ) + J C n v t ( z ) + e C ( z ) v ( z ) = 0 ∀ z = s + i t ∈ B ǫ , (3.5)where e C = Ψ − · (cid:0) Ψ s + J Ψ t + C Ψ) ∈ L p ( B ǫ ; End R C n ) . Thus, we have reduced the problem to the case J = J C n .(2) Let e C ± = ( e C ∓ J C n e CJ C n ) be the C -linear and C -antilinear parts of e C , i.e. e C ± J C n = ± J C n e C ± .With h· , ·i denoting the Hermitian inner-product on C n which is C -antilinear in the second input,define D ∈ L ∞ ( B ǫ ; End R C n ) , D ( z ) w = ( | v ( z ) | − h v ( z ) , w i v ( z ) , if v ( z ) = 0;0 , otherwise; A = e C + + e C − D .
Since DJ C n = − J C n D and Dv = v , A ∈ L p ( B ǫ ; End C C n ) and Av = e Cv . Thus, by (3.5), v s + J C n v t + Av = 0 . The claim now follows from Lemma 3.5.
Corollary 3.6.
Suppose n ∈ Z + , ǫ ∈ R + , J is a smooth almost complex structure on C n with J = J C n , and u : B ǫ −→ C n is a J -holomorphic map with u (0) = 0 . Then, there exist δ ∈ (0 , ǫ ) , C ∈ R + , Φ ∈ C ( B δ ; GL n R ) , and a J C n -holomorphic map σ : B δ −→ C n such that Φ is smoothon B ∗ δ , σ (0) = 0 , Φ(0) = Id C n , J ( u ( z ))Φ( z ) = Φ( z ) J C n , u ( z ) = Φ( z ) σ ( z ) , (cid:12)(cid:12) d z Φ (cid:12)(cid:12) ≤ C ∀ z ∈ B ∗ δ . Proof.
We can assume that u is not identically 0 on some neighborhood of 0 ∈ B ǫ . Similarly to (1)in the proof of Proposition 3.1, there existsΨ ∈ C ∞ ( C n ; GL n R ) s.t. Ψ(0) = Id C n , J ( x )Ψ( x ) = Ψ( x ) J C n ∀ x ∈ C n . Let v ( z ) = Ψ( u ( z )) − u ( z ). By Corollary 3.3, we can choose complex linear coordinates on C n so that v ( z ) = (cid:0) f ( z ) , g ( z ) (cid:1) h ( z ) ∈ C ⊕ C n − ∀ z ∈ B ǫ ′ for some ǫ ′ ∈ (0 , ǫ ), holomorphic function h on B ǫ ′ with h (0) = 0, and continuous functions f and g on B ǫ ′ with f (0) = 1 and g (0) = 0. By Lemma 3.7 below applied with f above and with eachcomponent of g separately, there exists δ ∈ (0 , ǫ ′ ) so that the functionΦ : B δ −→ GL n R , Φ( z ) = Ψ (cid:0) u ( z ) (cid:1) (cid:18) f ( z ) 0 g ( z ) 1 (cid:19) , is continuous on B δ and smooth on B δ − | d z Φ | uniformly bounded on B δ −
0. Taking σ ( z ) = ( h ( z ) , emma 3.7. Suppose ǫ ∈ R + , and f, h : B ǫ −→ C are continuous functions such that h is holomor-phic, h ( z ) = 0 for some z ∈ B ǫ , and the function B ǫ −→ C , z −→ f ( z ) h ( z ) , (3.6) is smooth. Then there exist δ ∈ (0 , ǫ ) and C ∈ R + such that f is differentiable on B ǫ − and (cid:12)(cid:12) d z f (cid:12)(cid:12) ≤ C ∀ z ∈ B δ − . (3.7) Proof.
After a holomorphic change of coordinate on B δ ⊂ B ǫ , we can assume that h ( z ) = z ℓ forsome ℓ ∈ Z ≥ . Define g : B δ −→ C , g ( z ) = f ( z ) z ℓ − f (0) z ℓ . By Taylor’s Theorem and the smoothness of the function (3.6), there exists
C > g satisfies (cid:12)(cid:12) g ( z ) (cid:12)(cid:12) ≤ C | z | ℓ +1 ∀ z ∈ B δ . Dividing g by z ℓ , we thus obtain (3.7). Remark 3.8.
Corollary 3.6 refines the conclusion of Proposition 3.1 for J -holomorphic maps.In contrast to the output (Φ , σ ) of Proposition 3.1, the output of Corollary 3.6 does not dependcontinuously on the input u with respect to the L p -norms. This makes Corollary 3.6 less suitablefor applications in settings involving families of J -holomorphic maps. J -holomorphic maps We now obtain three corollaries from Proposition 3.1. They underpin important geometric state-ments established later in these notes, such as Propositions 3.12 and 4.11 and Lemma 5.4.
Corollary 3.9 (Unique Continuation) . Suppose ( X, J ) is an almost complex manifold, (Σ , j ) is aconnected Riemann surface, and u, u ′ : (Σ , j ) −→ ( X, J ) are J -holomorphic maps. If u and u ′ agree to infinite order at z ∈ Σ , then u ′ = u ′ .Proof. Since the subset of the points of Σ at which u and u ′ agree is closed to infinite order, it isenough to show that u = u ′ on some neighborhood of z . By the continuity of u , we can assumethat X = C n , Σ = B , z = 0, and u (0) , u ′ (0) = 0. Let w = u ′ − u : B ǫ −→ C n . Since J is C , J ( x + y ) = J ( x ) + Z d J ( x + ty )d t d t = J ( x ) + n X i =1 y i Z ∂J∂y i (cid:12)(cid:12)(cid:12)(cid:12) x + ty d t . (3.8)Since u and u ′ are J -holomorphic, (3.8) implies that ∂ s w + J (cid:0) u ( z ) (cid:1) ∂ t w + C ( z ) w ( z ) = 0 , where C ∈ L p (cid:0) B ; End R C n (cid:1) ,C ( z ) y = n X i =1 y i (cid:18) Z ∂J∂y i (cid:12)(cid:12)(cid:12)(cid:12) v ( z )+ tw ( z ) d t (cid:19) ∂ t w | z .
16y Proposition 3.1, there thus exist δ ∈ (0 , ∈ L p ( B δ ; GL n R ), and holomorphic map e w : B δ −→ C n such that w ( z ) = Φ( z ) e w ( z ) ∀ z ∈ B δ . Since w vanishes to infinite order at 0, it follows that e w ( z ) = 0 for all z ∈ B δ (otherwise, w wouldsatisfy the conclusion of Corollary 3.3) and thus w ( z ) = 0 for all z ∈ B δ . Corollary 3.10.
Suppose ( X, J ) is an almost complex manifold, u, u ′ : (Σ , j ) , (Σ ′ , j ′ ) −→ ( X, J ) are J -holomorphic maps, z ∈ Σ is such that d z u = 0 , and z ′ ∈ Σ ′ is such that u ′ ( z ′ ) = u ( z ) . Ifthere exist sequences z i ∈ Σ − z and z ′ i ∈ Σ ′ − z ′ such that lim i −→∞ z i = z , lim i −→∞ z ′ i = z ′ , and u ( z i ) = u ′ ( z i ) ∀ i ∈ Z + , then there exists a holomorphic map σ : U ′ −→ Σ from a neighborhood of z ′ in Σ ′ such that σ ( z ′ ) = z and u ′ | U ′ = u ◦ σ .Proof. It can be assumed that (Σ , j , z ) , (Σ ′ , j ′ , z ′ ) = ( B , j , B ⊂ C is the unit ball withthe standard complex structure. Since d z u = 0 and u is J -holomorphic, u is an embedding near0 ∈ B and so is a slice in a coordinate system. Thus, we can assume that u, u ′ ≡ ( v, w ) : ( B , −→ ( C × C n − , , u ( z ) = ( z, ∈ C × C n − , and u, u ′ are J -holomorphic with respect to some almost complex structure J ( x, y ) = (cid:18) J ( x, y ) J ( x, y ) J ( x, y ) J ( x, y ) (cid:19) : C × C n − −→ C × C n − , ( x, y ) ∈ C × C n − . Since J is C , J ij ( x, y ) = J ij ( x,
0) + Z d J ij ( x, ty )d t d t = J ij ( x,
0) + n − X i =1 y i Z ∂J ij ∂y i (cid:12)(cid:12)(cid:12)(cid:12) ( x,ty ) d t . (3.9)Since u is J -holomorphic, J ( x,
0) = 0 , J ( x, = − Id ∀ x ∈ B ⊂ C . (3.10)Since u ′ is J -holomorphic, ∂ s w + J (cid:0) v ( z ) , w ( z ) (cid:1) ∂ t w + J (cid:0) v ( z ) , w ( z ) (cid:1) ∂ t v = 0 . Combining this with (3.9) and the first equation in (3.10), we find that ∂ s w + J (cid:0) v ( z ) , (cid:1) ∂ t w + C ( z ) w ( z ) = 0 , where C ∈ L p (cid:0) B ; End R C n − (cid:1) ,C ( z ) y = n − X i =1 y i (cid:18) Z ∂J ∂y i (cid:12)(cid:12)(cid:12)(cid:12) ( v ( z ) ,tw ( z )) d t (cid:19) ∂ t w | z + (cid:18) Z ∂J ∂y i (cid:12)(cid:12)(cid:12)(cid:12) ( v ( z ) ,tw ( z )) d t (cid:19) ∂ t v | z ! . By Proposition 3.1 and the second identity in (3.10), there thus exist δ ∈ (0 , ∈ L p ( B δ ; GL n − R ),and holomorphic map e w : B δ −→ C n − such that w ( z ) = Φ( z ) e w ( z ) ∀ z ∈ B δ . Since u ′ ( z ′ i ) = u ( z i ), e w ( z ′ i ) = 0 for all i ∈ Z + . Since z ′ i −→ z ′ i = 0, it follows that w = 0. Thisimplies the claim with U ′ = B δ and σ = v . 17 orollary 3.11. Let ( X, J ) be an almost complex manifold with a Riemannian metric g and x ∈ X be such that g is compatible with J at x . If u : Σ −→ X is a J -holomorphic map from a compactRiemann surface with boundary, then lim δ −→ E g ( u ; u − ( B gδ ( x ))) πδ = ord x u . Proof.
By the continuity of u , we can assume that X = C n , J agrees with the standard complexstructure J C n at the origin, g agrees with the standard metric g C n at the origin, Σ = B R for some R ∈ R + , and u (0) = 0. In particular, there exists C ≥ (cid:12)(cid:12) J x − J C n (cid:12)(cid:12) ≤ C | x | , (cid:12)(cid:12) g x − g C n (cid:12)(cid:12) ≤ C | x | ∀ x ∈ C n s.t. | x | ≤ , (3.11)where | · | denotes the usual norm of x (i.e. the distance to the origin with respect to g C n ).Let ℓ ≡ ord u and α ∈ C n − − ∈ B R is the origin in the domain of u .Thus, there exist ǫ ∈ (0 ,
1) and C ∈ R + such that u ( z ) = α · (cid:0) z ℓ + f ( z ) (cid:1) , (cid:12)(cid:12) f ( z ) (cid:12)(cid:12) ≤ C | z | ℓ +1 ∀ z ∈ B ǫ . (3.12)Let z = s + i t as before. By (3.12), there exists C ∈ R + such that u s ( z ) = α · (cid:0) ℓz ℓ − + f s ( z ) (cid:1) , u s ( z ) = α · (cid:0) ℓ i z ℓ − + f t ( z ) (cid:1) , (cid:12)(cid:12) f s ( z ) (cid:12)(cid:12) , (cid:12)(cid:12) f t ( z ) (cid:12)(cid:12) ≤ C | z | ℓ ∀ z ∈ B ǫ . (3.13)We can also assume that the three constants C in (3.11), (3.12), and (3.13) are the same, C ≥ C α ǫ ≡ ( C + C | α | + C | α | (cid:1) ǫ ≤ , and | u ( z ) | ≤ z ∈ B ǫ . By (3.11)-(3.13), (cid:12)(cid:12)(cid:12)(cid:12) | u ( z ) | g | α || z | ℓ − (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) | u s ( z ) | g | α | ℓ | z | ℓ − − (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) | u t ( z ) | g | α | ℓ | z | ℓ − − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | z | + C | α || z | ℓ + C | α || z | ℓ +1 ≤ C α | z | ∀ z ∈ B ǫ , (3.14)where | · | g denotes the distance to the origin in C n with respect to the metric g and the corre-sponding norm on T C n .Given r ∈ (0 , δ r ∈ (0 , ǫ ) be such that C α (cid:18) δ r (1 − r ) | α | (cid:19) /ℓ ≤ r . (3.15)For any δ ∈ [0 , δ r ], (3.14) and (3.15) give | z | ≤ (cid:18) δ (1+ r ) | α | (cid:19) /ℓ = ⇒ u ( z ) ∈ B gδ (0) ,u ( z ) ∈ B gδ (0) = ⇒ | z | ≤ (cid:18) δ (1 − r ) | α | (cid:19) /ℓ , | z | ≤ (cid:18) δ (1 − r ) | α | (cid:19) /ℓ = ⇒ − r ≤ | u s ( z ) | g | α | ℓ | z | ℓ − , | u t ( z ) | g | α | ℓ | z | ℓ − ≤ r. Z | z |≤ (cid:16) δ (1+ r ) | α | (cid:17) ℓ (1 − r ) (cid:0) | α | ℓ | z | ℓ − (cid:1) ≤ Z u − ( B gδ (0)) (cid:0) | u s | g + | u t | g (cid:1) ≤ Z | z |≤ (cid:16) δ (1 − r ) | α | (cid:17) ℓ (1+ r ) (cid:0) | α | ℓ | z | ℓ − (cid:1) . Evaluating the outer integrals, we find that (cid:18) − r r (cid:19) ℓπδ ≤ E g (cid:0) u ; u − ( B gδ (0)) (cid:1) ≤ (cid:18) r − r (cid:19) ℓπδ . These inequalities hold for all r ∈ (0 ,
1) and δ ∈ (0 , δ r ); the claim is obtained by sending r −→ Proposition 3.12 below is a key step in the continuity part of the proof of the
Removal of Singularity
Proposition 5.1. The precise nature of the lower energy bound on the right hand-side of (3.16)does not matter, as long as it is positive for δ > Proposition 3.12 (Monotonicity Lemma) . If ( X, J ) is an almost complex manifold and g is aRiemannian metric on X compatible with J , there exists a continuous function C g,J : X −→ R + with the following property. If (Σ , j ) is a compact Riemann surface with boundary, u : Σ −→ X is a J -holomorphic map, x ∈ X , and δ ∈ R + are such that u ( ∂ Σ) ∩ B gδ ( x ) = ∅ , then E g ( u ) ≥ (cid:0) ord x u (cid:1) πδ (1+ C g,J ( x ) δ ) . (3.16) If ω ( · , · ) ≡ g ( J · , · ) is a symplectic form on X , then the above fraction can be replaced by the product πδ e − C g,J ( x ) δ . According to this proposition, “completely getting out” of the ball B δ ( x ) via a J -holomorphic maprequires an energy bounded below by a little less than πδ . Thus, the L -norm of a J -holomorphicmap u exerts some control over the C -norm of u . If p >
2, the L p -norm of any smooth map f from a two-dimensional manifold controls the C -norm of f . However, this is not the case of the L -norm, as illustrated by the example of [12, Lemma 10.4.1]: the function f ǫ : R −→ [0 , , f ǫ ( z ) = , if | z | ≤ ǫ ; ln | z | ln ǫ , if ǫ ≤ | z | ≤ , if | z | ≥ ǫ ∈ (0 ,
1) is continuous and satisfies Z R | d f ǫ | g = − π ln ǫ . It is arbitrarily close in the L -norm to a smooth function e f ǫ . Thus, it is possible to “completelyget out” of B gδ ( x ) using a smooth function with arbitrarily small energy ( e f δ does this for the ball B (1) in R ). 19y (2.7), the holomorphic maps are the local minima of the functional C ∞ (Σ; X ) −→ R , f −→ E g ( f ) − Z Σ f ∗ ω J , for every compact Riemann surface (Σ , j ) without boundary. This fact underlines Lemma 3.16,the key ingredient in the proof of the Monotonicity Lemma. Lemma 3.16 implies that the ratio of E g ( u ; u − ( B gδ ( x ))) and the fraction on the right-hand side (3.16) is a non-decreasing function of δ ,as long as u ( ∂ Σ) ∩ B gδ ( x ) = ∅ . By Corollary 3.11, this ratio approaches ord x u as δ approaches 0.These two statements imply Proposition 3.12.We first make some general Riemannian geometry observations. Let ( X, g ) be a Riemannianmanifold. Denote by exp : W g −→ X , the exponential map from a neighborhood of X in T X withrespect to the Levi-Civita connection ∇ of g . For each v ∈ T X , we denote by γ v : [0 , −→ X, γ v ( τ ) = exp x ( τ v ) , the geodesic with γ ′ v (0) = v . Let r g : X −→ R + and d g : X × X −→ R ≥ be the injectivity radius of exp and the distance function. For each x ∈ X , define ζ x ∈ Γ (cid:0) B gr g ( x ) ( x ); T X (cid:1) by exp y (cid:0) ζ x ( y ) (cid:1) = x, g (cid:0) ζ x ( y ) , ζ x ( y ) (cid:1) < r g ( x ) ∀ y ∈ B gr g ( x ) ( x ) . Lemma 3.13.
Let ( X, g ) be a Riemannian manifold and x ∈ X . If α : ( − ǫ, ǫ ) −→ X is a smoothcurve such that α (0) ∈ B gr g ( x ) ( x ) , then
12 dd τ d g (cid:0) x, α ( τ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = − g (cid:0) α ′ (0) , ζ x ( α (0)) (cid:1) . Proof. If β ( τ ) = exp − x α ( τ ), then12 dd τ d g (cid:0) x, α ( τ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = 12 dd τ | β ( τ ) | (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = g (cid:0) β ′ (0) , β (0) (cid:1) . By Gauss’s Lemma, g (cid:0) β ′ (0) , β (0) (cid:1) = g (cid:0) { d β (0) exp x } ( β ′ (0)) , { d β (0) exp x } ( β (0)) (cid:1) = g (cid:0) α ′ (0) , − ζ x ( α (0)) (cid:1) . This establishes the claim.
Lemma 3.14. If ( X, g ) is a Riemannian manifold, there exists a continuous function C g : X −→ R + with the following property. If x ∈ X , v ∈ T x X with | v | g < r g ( x ) , and τ −→ J ( τ ) is a Jacobi vectorfield along the geodesic γ v with J (0) = 0 , then (cid:12)(cid:12) J ′ (1) − J (1) (cid:12)(cid:12) g ≤ C g ( x ) | v | g (cid:12)(cid:12) J (1) (cid:12)(cid:12) g . Proof.
Let R g be the Riemann curvature tensor of g and f ( τ ) = | τ J ′ ( τ ) − J ( τ ) | g . Then, f (0) = 0and f ( τ ) f ′ ( τ ) = 12 dd τ f ( τ ) = g (cid:0) τ J ′′ ( τ ) , τ J ′ ( τ ) − J ( τ ) (cid:1) = τ g (cid:0) R ( γ ′ ( τ ) , J ( τ )) γ ′ ( τ ) , τ J ′ ( τ ) − J ( τ ) (cid:1) ≤ C g ( x ) | v | g | J ( τ ) | g τ f ( τ ) . C g is sufficiently large, then | J ( τ ) | g ≤ C g ( x ) | J (1) | g . Thus, f ( τ ) f ′ ( τ ) ≤ C g ( x ) | v | g | J v ( τ ) | g τ f ( τ ) ≤ C g ( x ) | v | g | J (1) | g τ f ( τ ) , f ′ ( τ ) ≤ C g ( x ) | v | g | J (1) | g τ. The claim follows from the last inequality.
Corollary 3.15. If ( X, g ) is a Riemannian manifold, there exists a continuous function C g : X −→ R + with the following property. If x ∈ X , then (cid:12)(cid:12) ∇ w ζ x | y + w (cid:12)(cid:12) g ≤ C g ( x ) d g ( x, y ) | w | g ∀ w ∈ T y X, y ∈ B gr g ( x ) / ( x ) . Proof.
Let τ −→ u ( s, τ ) be a family of geodesics such that u ( s,
0) = x, u (0 ,
1) = y, dd s u ( s, (cid:12)(cid:12)(cid:12)(cid:12) s =0 = w. Since τ −→ u ( s, τ ) is a geodesic,dd τ u ( s, τ ) (cid:12)(cid:12)(cid:12)(cid:12) τ =1 = (cid:8) d u τ ( s, exp x (cid:9)(cid:0) u τ ( s, (cid:1) = − ζ x (cid:0) u ( s, (cid:1) , Dd τ d u ( s, τ )d s (cid:12)(cid:12)(cid:12)(cid:12) ( s,τ )=(0 , = Dd s d u ( s, τ )d τ (cid:12)(cid:12)(cid:12)(cid:12) ( s,τ )=(0 , = −∇ w ζ x | y . Furthermore, J ( τ ) ≡ dd s u ( s, τ ) (cid:12)(cid:12) s =0 is a Jacobi vector field along the geodesic τ −→ u (0 , τ ) with J (0) = 0 , J (1) = w, J ′ (1) = Dd τ d u ( s, τ )d s (cid:12)(cid:12)(cid:12)(cid:12) ( s,τ )=(0 , = −∇ w ζ x | y . Thus, the claim follows from Lemma 3.14.
Lemma 3.16.
Suppose ( X, ω ) is a symplectic manifold, J is an almost complex structure on X tamed by ω , and ∇ is the Levi-Civita connection of the metric g J . If (Σ , j ) is a compact Riemannsurface with boundary and u : Σ −→ X is a J -holomorphic map, then Z Σ g J (cid:0) d u ⊗ j ∇ ξ (cid:1) = Z Σ (cid:0) u ∗ {∇ ξ ω J } + ω J (d u ∧ j ∇ ξ ) (cid:1) ∀ ξ ∈ Γ(Σ; u ∗ T X ) s.t. ξ | ∂ Σ = 0 . Proof.
For τ ∈ R sufficiently close to 0, define u τ : Σ −→ X, u τ ( z ) = exp u ( z ) ( τ ξ ( z )) . Since ξ | ∂ Σ = 0, u τ | ∂ Σ = u | ∂ Σ . Denote by b Σ the closed oriented surface obtained by gluing two copiesof Σ along the common boundary and reversing the orientation on the second copy. Let b u τ : b Σ −→ X be the map restricting to u τ on the first copy of Σ and to u on the second.By (2.7), E ( τ ) ≡ E g J ( u τ ) − Z Σ u ∗ τ ω J − E g J ( u ) = Z b Σ b u ∗ τ ω + 2 Z Σ g J (cid:0) ¯ ∂u τ ⊗ j ¯ ∂u τ (cid:1) ≥ ∀ τ. ω is closed and b u ∗ represents the zero class in H ( X ; Z ), the first integral on the right-handside above vanishes. Thus, the function τ −→ E ( τ ) is minimized at τ = 0 (when it equals 0) and so0 = E ′ (0) = dd τ (cid:18) E g J ( u τ ) − Z Σ u ∗ τ ω J (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) τ =0 = dd τ (cid:18) Z Σ g J (d u τ ⊗ j d u τ ) − Z Σ u ∗ τ ω J (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) τ =0 ; (3.17)the last equality above uses the definition of E ( u τ ) in (2.5).Let z = s + i t be a local coordinate on (Σ , j ). Since ∇ is torsion-free, D d τ ( u τ ) s (cid:12)(cid:12)(cid:12) τ =0 ≡ D d τ d u τ d s (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = D d s d u τ d τ (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = D d s ξ ≡ ∇ s ξ, D d τ ( u τ ) t (cid:12)(cid:12)(cid:12) τ =0 = ∇ t ξ . Since ∇ is also g -compatible,12 dd τ g J (d u τ ⊗ j d u τ ) (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = (cid:18) g J (cid:18) u s , D d τ ( u τ ) s (cid:12)(cid:12)(cid:12) τ =0 (cid:19) + g J (cid:18) u t , D d τ ( u τ ) t (cid:12)(cid:12)(cid:12) τ =0 (cid:19)(cid:19) d s ∧ d t = g J ( u s , ∇ s ξ ) + g J ( u t , ∇ t ξ ) = g J (cid:0) d u ⊗ j ∇ ξ (cid:1) , dd τ u ∗ τ ω J (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = (cid:18)(cid:8) ∇ ξ ω J (cid:9) ( u s , u t ) + ω J (cid:18) D d τ ( u τ ) s (cid:12)(cid:12)(cid:12) τ =0 , u t (cid:19) + ω J (cid:18) u s , D d τ ( u τ ) t (cid:12)(cid:12)(cid:12) τ =0 (cid:19)(cid:19) d s ∧ d t = u ∗ {∇ ξ ω J } + ω J (cid:0) d u ∧ j ∇ ξ (cid:1) . Combining this with (3.17), we obtain the claim.
Proof of Proposition 3.12 . Let δ g : X −→ R + be a continuous function such that for every x ∈ X there exists a symplectic form ω x on B g δ g ( x ) ( x ) so that J is tamed by ω x on B g δ g ( x ) ( x )and compatible with ω x at x . We assume that 2 δ g ( x ) ≤ r g ( x ) for every x ∈ X . It is sufficient toestablish the proposition for each x ∈ X and each δ ≤ δ g ( x ) under the assumption that the metric g is determined by J and ω x on B gδ g ( x ) ( x ).Choose a C ∞ -function η : R −→ [0 ,
1] such that η ( τ ) = ( , if τ ≤ ;0 , if τ ≥ η ′ ( τ ) ≤ . (3.18)For a compact Riemann surface with boundary (Σ , j ), a smooth map u : Σ −→ X , x ∈ X , and δ ∈ R + ,define η u,x,δ ∈ C ∞ (Σ; R ) , η u,x,δ ( z ) = η (cid:18) d g ( x, u ( z )) δ (cid:19) ,E u,x,η ( δ ) = 12 Z Σ η u,x,δ ( z ) g (cid:0) d u ⊗ j d u (cid:1) , E u,x ( δ ) = E g (cid:0) u ; u − ( B gδ ( x )) (cid:1) . We show in the remainder of this proof that there exists a continuous function C g,J : X −→ R + such that − δE ′ u,x,η ( δ ) + 2 E u,x,η ( δ ) ≤ C g,J ( x ) δE u,x,η ( δ ) + C g,J ( x ) δ E ′ u,x,η ( δ ) (3.19)22or every compact Riemann surface with boundary (Σ , j ), J -holomorphic map u : Σ −→ X , and δ ∈ (0 , δ g ( x )) such that u ( ∂ Σ) ∩ B gδ ( x ) = ∅ . This inequality is equivalent to (cid:18) E u,x,η ( δ ) (cid:30) δ (1+ C g,J ( x ) δ ) (cid:19) ′ ≥ . By Lebesgue’s Dominated Convergence Theorem, E u,x,η ( δ ) approaches E u,x ( δ ) from below as η approaches the characteristic function χ ( −∞ , of ( −∞ , δ −→ E u,x ( δ ) (cid:30) δ (1+ C g,J ( x ) δ ) is non-decreasing as long as u ( ∂ Σ) ∩ B gδ ( x ) = ∅ . By Corollary 3.11,lim δ −→ (cid:18) E u,x ( δ ) (cid:30) δ (1+ C g,J ( x ) δ ) (cid:19) = lim δ −→ E u,x ( δ ) δ = (cid:0) ord x u (cid:1) π. This implies the first claim.Fix x ∈ X . We note that E ′ u,x,η ( δ ) = − Z Σ η ′ (cid:18) d g ( x, u ( z )) δ (cid:19) d g ( x, u ( z )) δ g (cid:0) d u ⊗ j d u (cid:1) . (3.20)For a compact Riemann surface with boundary (Σ , j ), a smooth map u : Σ −→ X , and δ ∈ (0 , δ g ( x )),let ξ u,x,δ ∈ Γ(Σ; u ∗ T X ) , ξ u,x,δ ( z ) = − η u,x,δ ( z ) ζ x (cid:0) u ( z ) (cid:1) ;the vanishing assumption in (3.18) implies that ξ u,x,δ is well-defined. If u ( ∂ Σ) ∩ B gδ ( x ) = ∅ , then ξ u,x,δ | ∂ Σ = 0. By Lemma 3.13, ∇ ξ u,x,δ | z = η ′ (cid:18) d g ( x, u ( z )) δ (cid:19) δ d g ( x, u ( z )) g (cid:0) d z u, ζ x ( u ( z )) (cid:1) ζ x ( u ( z )) − η u,x,δ ( z ) ∇ ζ x ◦ d z u. (3.21)Along with Corollary 3.15, (3.20), and the last assumption in (3.18), this implies that Z Σ d g ( x, u ( z )) (cid:12)(cid:12) g (d u ⊗ j ∇ ξ u,x,δ ) (cid:12)(cid:12) ≤ δ E ′ u,x,η ( δ ) + 2 (cid:0) C g ( x ) δ (cid:1) δE u,x,η ( δ ) . (3.22)By the ω x -compatibility assumption on J at x , there exists a continuous function C : X −→ R + such that Z Σ (cid:12)(cid:12) ( ω x ) J (d u ∧ j ∇ ξ u,x,δ ) (cid:12)(cid:12) ≤ C ( x ) Z Σ d g (cid:0) x, u ( z ) (cid:1)(cid:12)(cid:12) g (d u ⊗ j ∇ ξ u,x,δ ) (cid:12)(cid:12) for all u and δ as above. Along with this, Lemma 3.16 implies that there exists a continuousfunction C : X −→ R + such that (cid:12)(cid:12)(cid:12)(cid:12) Z Σ g (cid:0) d u ⊗ j ∇ ξ u,x,δ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( x ) Z Σ (cid:0) g (cid:0) d u ⊗ j d u (cid:1) | ξ u,x,δ | + d g ( x, u ( z )) (cid:12)(cid:12) g (d u ⊗ j ∇ ξ u,x,δ ) (cid:12)(cid:12)(cid:1) for every compact Riemann surface with boundary (Σ , j ), J -holomorphic map u : Σ −→ X , and δ ∈ (0 , δ g ( x )) such that u ( ∂ Σ) ∩ B gδ ( x ) = ∅ . Combining this with (3.22), we conclude that there existsa continuous function C : X −→ R + such that (cid:12)(cid:12)(cid:12)(cid:12) Z Σ g (cid:0) d u ⊗ j ∇ ξ u,x,δ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( x ) (cid:0) δE u,x,η ( δ )+ δ E ′ u,x,η ( δ ) (cid:1) (3.23)23or all u and δ as above.Suppose (Σ , j ) is a compact Riemann surface with boundary, u : Σ −→ X is a smooth map, and δ ∈ (0 , δ g ( x )). Let z = s + i t be a coordinate on (Σ , j ). By (3.21), g (cid:0) u s , ∇ s ξ u,x,δ (cid:1) = η ′ (cid:18) d g ( x, u ( z )) δ (cid:19) δ d g ( x, u ( z )) g (cid:0) u s , ζ x ( u ( z )) (cid:1) + η u,x,δ ( z ) g (cid:0) u s , ∇ s ( − ζ x ) | z (cid:1) . (3.24)By Corollary 3.15, | u s | ≤ g (cid:0) u s , ∇ s ( − ζ x ) | z (cid:1) + C g ( x ) d g ( x, u ( z )) | u s | ∀ z ∈ u − (cid:0) B gδ g ( x ) ( x ) (cid:1) . (3.25)If u is J -holomorphic, then | u s | = | u t | , h u s , u t i = 0, and12 (cid:0) | u s | + | u t | (cid:1) d g ( x, u ( z )) = | u s | | ζ x ( u ( z )) | ≥ g (cid:0) u s , ζ x ( u ( z )) (cid:1) + g (cid:0) u t , ζ x ( u ( z )) (cid:1) . (3.26)Since η ′ ≤
0, (3.24)-(3.26) give12 η ′ (cid:18) d g ( x, u ( z )) δ (cid:19) d g ( x, u ( z )) δ (cid:0) | u s | + | u t | (cid:1) + η u,x,δ ( z ) (cid:0) | u s | + | u t | (cid:1) ≤ g (cid:0) u s , ∇ s ξ u,x,δ (cid:1) + g (cid:0) u t , ∇ t ξ u,x,δ (cid:1) + C g ( x ) η u,x,δ ( z ) d g ( x, u ( z )) (cid:0) | u s | + | u t | (cid:1) . (3.27)Along with (3.20), this implies that − δE ′ u,x,η ( δ ) + 2 E u,x,η ( δ ) ≤ Z Σ g (cid:0) d u ⊗ j ∇ ξ u,x,δ (cid:1) + 2 C g ( x ) δ E u,x,η ( δ ) (3.28)for every compact Riemann surface with boundary (Σ , j ), J -holomorphic map u : Σ −→ X , and δ ∈ (0 , δ g ( x )). Combining this inequality with (3.23), we obtain (3.19).Suppose ω ≡ g ( J · , · ) is a symplectic form on X . By Lemma 3.16, the left-hand side of (3.23) thenvanishes. From (3.28), we thus obtain − δE ′ u,x,η ( δ ) + 2 E u,x,η ( δ ) ≤ C g,J ( x ) δ E u,x,η ( δ ) . The reasoning below (3.19) now yields the second claim.
We now move to properties of J -holomorphic maps u from Riemann surfaces (Σ , j ) into almostcomplex manifolds ( X, J ) that are of a more global nature. They generally concern the distributionof the energy of such a map over its domain and are consequences of the
Mean Value Inequality for J -holomorphic maps. These fairly technical properties lead to geometric conclusions such asPropositions 4.3 and 5.1. 24 .1 Statement and proof According to Cauchy’s Integral Formula, a holomorphic map u : B R −→ C n satisfies u ′ (0) = 12 π i I | z | = r u ( z ) z d z ∀ r ∈ (0 , R ) . This immediately implies that a bounded holomorphic function defined on all of C is constant. TheMean Value Inequality of Proposition 4.1 bounds the norms of the differentials of J -holomorphicmaps of sufficiently small energy away from the boundary of the domain “uniformly” by their L -norms. In general, one would not expect the value of a function to be bounded by its integral.The Mean Value Inequality implies that a J -holomorphic map which is defined on all of C and hassufficiently small energy is in fact constant; see Corollary 4.2. Proposition 4.1 (Mean Value Inequality) . If ( X, J ) is an almost complex manifold and g is aRiemannian metric on X compatible with J , there exists a continuous function ~ J,g : X × R −→ R + with the following property. If u : B R −→ X is a J -holomorphic map such that u ( B R ) ⊂ B gr ( x ) and E g ( u ) < ~ J,g ( x, r ) for some x ∈ X and r ∈ R , then (cid:12)(cid:12) d u (cid:12)(cid:12) g < πR E g ( u ) . (4.1) Proof.
Let φ ( z ) = | d z u | g . By Lemma 4.7 below, ∆ φ ≥ − A J,g φ with A J,g : X × R −→ R + determinedby ( X, J, g ). The claim with ~ J,g = π/ A J,g thus follows from Proposition 4.6.
Corollary 4.2 (Lower Energy Bound) . If ( X, J ) is a compact almost complex manifold and g is aRiemannian metric on X , then there exists ~ J,g ∈ R + such that E g ( u ) ≥ ~ J,g for every non-constant J -holomorphic map u : S −→ X .Proof. By the compactness of X , we can assume that g is compatible with J . Let ~ J,g > ~ J,g in the statement of Proposition 4.1 on the compact space X × [0 , diam g ( X )]. If u : S −→ X is J -holomorphic map with E g ( u ) < ~ J,g , (cid:12)(cid:12) d z u (cid:12)(cid:12) g < πR E g (cid:0) u ; B R ( z ) (cid:1) ≤ πR E g ( u ) ∀ z ∈ C , R ∈ R + by Proposition 4.1, since B R ( z ) ⊂ C as Riemann surfaces. Thus, d z u = 0 for all z ∈ C , and so u isconstant.If φ : U −→ R is a C -function on an open subset of R , let∆ φ = ∂ φ∂s + ∂ φ∂t ≡ φ ss + φ tt denote the Laplacian of φ . Exercise 4.3.
Show that in the polar coordinates ( r, θ ) on R ,∆ φ = φ rr + r − φ r + r − φ θθ . (4.2) Lemma 4.4. If φ : B R −→ R is C , then πR φ (0) = − R Z ( r,θ ) ∈ B R (ln R − ln r )∆ φ + Z ∂B R φ . (4.3)25 roof. By Stokes’ Theorem applied to φ d θ on B R − B δ , I ∂B R φ d θ − I ∂B δ φ d θ = Z B R − B δ φ r d r ∧ d θ = Z π Z Rδ ( rφ r ) r − d r d θ = Z π (ln R − ln δ ) δ φ r ( δ, θ )d θ + Z π Z Rδ (ln R − ln r )( φ rr + r − φ r ) r d r d θ ;the last equality above is obtained by applying integration by parts to the functions ln r − ln R and rφ r . Sending δ −→ R Z ∂B R φ − π φ (0) = 0 + Z ( r,θ ) ∈ B R (ln R − ln r )∆ φ , which is equivalent to (4.3). Corollary 4.5. If φ : B R −→ R is C and ∆ φ ≥ − C for some C ∈ R + , then φ (0) ≤ CR + 1 πR Z B R φ . (4.4) Proof.
By (4.3),2 πr φ (0) ≤ Cr Z π Z r (ln r − ln ρ ) ρ d ρ d θ + Z ∂B r φ = Cr · π · r Z ∂B r φ ∀ r ∈ (0 , R ) . Integrating the above in r ∈ (0 , R ), we obtain2 πφ (0) · R ≤ πC · R
16 + Z B R φ. This inequality is equivalent to (4.4).
Proposition 4.6. If φ : B R −→ R ≥ is C and there exists A ∈ R + such that ∆ φ ≥ − Aφ and Z B R φ < π A , then φ (0) ≤ πR Z B R φ . (4.5) Proof.
Replacing A by e A = R A and φ by e φ : B −→ R , e φ ( z ) = φ ( Rz ) , we can assume that R = 1, as well as that φ is defined on B .(1) Define f : [0 , −→ R ≥ by f ( r ) = (1 − r ) max B r φ . In particular, f (0) = φ (0) and f (1) = 0. Choose r ∗ ∈ [0 ,
1) and z ∗ ∈ B r ∗ such that f ( r ∗ ) = sup f and φ ( z ∗ ) = sup B r ∗ φ ≡ c ∗ . r ∗ δB δ ( z ∗ ) Figure 6: Setup for the proof of Proposition 4.6Let δ = (1 − r ∗ ) >
0; see Figure 6. Thus,sup B δ ( z ∗ ) φ ≤ sup B r ∗ + δ φ = f ( r ∗ + δ )(1 − ( r ∗ + δ )) ≤ f ( r ∗ ) (1 − r ∗ ) = 4 φ ( z ∗ ) = 4 c ∗ . In particular, ∆ φ ≥ − Aφ ≥ − Ac ∗ on B δ ( z ∗ ).(2) Using Corollary 4.5, we thus find that c ∗ = φ ( z ∗ ) ≤ · Ac ∗ · ρ + 1 πρ Z B ρ ( z ∗ ) φ ≤ Ac ∗ ρ + 1 πρ Z B φ ∀ ρ ∈ [0 , δ ] . (4.6)If 2 Ac ∗ δ ≤ , the ρ = δ case of the above inequality gives12 c ∗ ≤ πδ Z B φ , φ (0) = f (0) ≤ f ( r ∗ ) = 4 c ∗ · δ ≤ π Z B φ , as claimed. If 2 Ac ∗ δ ≥ , ρ ≡ (4 Ac ∗ ) − ≤ δ and (4.6) gives c ∗ ≤ Ac ∗ · Ac ∗ + 4 Ac ∗ π Z B φ . Thus, π A ≤ Z B φ , contrary to the assumption. Lemma 4.7. If ( X, J ) is an almost complex manifold and g is a Riemannian metric on X com-patible with J , there exists a continuous function A J,g : X × R −→ R + with the following property.If Ω ⊂ C is an open subset, u : Ω −→ X is a J -holomorphic map, and u (Ω) ⊂ B gr ( x ) for some x ∈ X and r ∈ R , then the function φ ( z ) ≡ | d z u | g satisfies ∆ φ ≥ − A J,g ( x, r ) φ .Proof. Let z = s + i t be the standard coordinate on C . Denote by u s and u t the s and t -partials of u ,respectively. Since u is J -holomorphic, i.e. u s = − J u t , and g is J -compatible, i.e. g ( J · , J · ) = g ( · , · ), | u s | g = | u t | g . Since the Levi-Civita connection ∇ of g is g -compatible and torsion-free,12 d d t | u s | g = |∇ t u s | g + (cid:10) ∇ t ∇ t u s , u t (cid:11) g = |∇ t u s | g + (cid:10) ∇ t ∇ s u t , u s (cid:11) g . (4.7)Similarly, 12 d d s | u t | g = (cid:12)(cid:12) ∇ s u t (cid:12)(cid:12) g + (cid:10) ∇ s ∇ t u s , u t (cid:11) g . (4.8)27ince u s = − J u t , h∇ s ∇ t u s , u t i g = − (cid:10) ∇ s ∇ t ( J u t ) , u t (cid:11) g = − (cid:10) J ∇ s ∇ t u t , u t (cid:11) g − (cid:10) ( ∇ s J ) ∇ t u t , u t (cid:11) g − (cid:10) ∇ s (( ∇ t J ) u t ) , u t (cid:11) g = − (cid:10) ∇ s ∇ t u t , u s (cid:11) g − (cid:10) ( ∇ s J ) ∇ t u t , u t (cid:11) g − (cid:10) ∇ s (( ∇ t J ) u t ) , u t (cid:11) g . (4.9)Putting (4.7)-(4.9), we find that12 ∆ φ = (cid:12)(cid:12) ∇ t u s (cid:12)(cid:12) g + (cid:12)(cid:12) ∇ s u t (cid:12)(cid:12) g + (cid:10) R g ( u t , u s ) u t , u s (cid:11) g − (cid:10) ( ∇ s J ) ∇ t u t , u t (cid:11) g − (cid:10) ∇ s (( ∇ t J ) u t ) , u t (cid:11) g , (4.10)where R g is the curvature tensor of the connection ∇ . Since u (Ω) ⊂ B gr ( x ), (cid:12)(cid:12) h R g ( u t , u s ) u t , u s i g (cid:12)(cid:12) ≤ C g ( x, r ) | u s | g | u t | g , (cid:12)(cid:12) h ( ∇ s J ) ∇ t u t , u t i g (cid:12)(cid:12) ≤ C J,g ( x, r ) | u s | g | u t | g (cid:12)(cid:12) ∇ t ( J u s ) (cid:12)(cid:12) g ≤ C J,g ( x, r ) | u s | g | u t | g (cid:0) | u s | g | u t | g + |∇ t u s | g (cid:1) ≤ ( C J,g ( x, r )+ C J,g ( x, r ) ) | u s | g | u t | g + |∇ t u s | g , (cid:12)(cid:12) h∇ s (( ∇ t J ) u t ) , u t i g (cid:12)(cid:12) ≤ C J,g ( x, r ) | u t | g (cid:0) | u s | g | u t | g + |∇ s u t | g (cid:1) ≤ C J,g ( x, r ) | u s | g | u t | g + C J,g ( x, r ) | u t | g + |∇ s u t | g . (4.11)Combining (4.10) and (4.11), we find that12 ∆ φ ≥ − C ( x, r ) (cid:0) | u s | g | u t | g + | u s | g | u t | g + | u t | g (cid:1) ≥ − C ( x, r ) φ , as claimed. J -holomorphic maps By Cauchy’s Integral Formula, a continuous extension of a holomorphic map u : B ∗ R −→ C n overthe origin is necessarily holomorphic. By Proposition 4.8 below, the same is the case for a J -holomorphic map u : B ∗ R −→ X of bounded energy. Proposition 4.8.
Let ( X, J ) be an almost complex manifold and g be a Riemannian metric on X .If R ∈ R + and u : B R −→ X is a continuous map such that u | B ∗ R is a J -holomorphic map and E g ( u ; B ∗ R ) < ∞ , then u is smooth and J -holomorphic on B R . For a smooth loop γ : S −→ X , define γ ′ ( θ ) = dd θ γ (cid:0) e i θ (cid:1) ∈ T γ (e i θ ) X and ℓ g ( γ ) = Z π (cid:12)(cid:12) γ ′ ( θ ) (cid:12)(cid:12) g d θ ∈ R ≥ to be the velocity of γ and the length of γ , respectively. Lemma 4.9 (Isoperimetric Inequality) . Let ( X, J, g ) , R , and u be as in Proposition 4.8 and γ r : S −→ X, γ r (cid:0) e i θ (cid:1) = u (cid:0) r e i θ (cid:1) ∀ r ∈ (0 , R ) . There exist δ ∈ (0 , R ) and C ∈ R + such that E g (cid:0) u ; B ∗ r (cid:1) ≤ Cℓ g ( γ r ) ∀ r ∈ (0 , δ ) . (4.12)28 ρ f γ ρ u | B r − B ρ γ ρ γ r γ r f γ r Figure 7: The maps from an annulus and two disks glued together to form the map F ρ ; r : S −→ X in the proof of Lemma 4.9 Proof.
Let exp be as above the statement of Lemma 3.13, δ g and ω x be as in the first two sentencesin the proof of Proposition 3.12, x = u (0) , δ = δ g ( x ) , ω = ω x , E : (0 , R ) −→ R , E ( r ) = E g ( u ; B ∗ r ) . We can assume that the metric g is determined by J and ω on B gδ ( x ).For a smooth loop γ : S −→ B gδ ( x ), define ξ γ : S −→ T x X by exp x ξ γ (cid:0) e i θ (cid:1) = γ (cid:0) e i θ (cid:1) , (cid:12)(cid:12) ξ γ (e i θ ) (cid:12)(cid:12) < δ ,f γ : B −→ X, f γ (cid:0) r e i θ (cid:1) = exp x (cid:0) rξ γ (e i θ ) (cid:1) . In particular, (cid:12)(cid:12) ∂ r f γ ( ρ e i θ ) (cid:12)(cid:12) g = (cid:12)(cid:12) ξ γ (e i θ ) (cid:12)(cid:12) g ≤ ℓ g ( γ ) / , (cid:12)(cid:12) r − ∂ θ f γ ( r e i θ ) (cid:12)(cid:12) g = (cid:12)(cid:12) d rξ γ (e i θ ) ( ξ ′ γ ( θ )) (cid:12)(cid:12) g ≤ C (cid:12)(cid:12) γ ′ ( θ ) (cid:12)(cid:12) g for some C ∈ R + determined by x . Thus, (cid:12)(cid:12)(cid:12)(cid:12) Z B f ∗ γ ω (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z π Z (cid:12)(cid:12) ∂ r f γ ( ρ e i θ ) (cid:12)(cid:12) g (cid:12)(cid:12) r − ∂ θ f γ ( r e i θ ) (cid:12)(cid:12) g r d r d θ ≤ C ′ ℓ g ( γ ) Z π Z r (cid:12)(cid:12) γ ′ ( θ ) (cid:12)(cid:12) g r d r d θ = 12 C ′ ℓ g ( γ ) (4.13)for some C, C ′ ∈ R + determined by x and ω .By Proposition 4.1 and the finiteness assumption on E ( u ; B ∗ R ), there exists δ ∈ (0 , R/
2) such that (cid:12)(cid:12) γ ′ r ( θ ) (cid:12)(cid:12) g ≡ (cid:12)(cid:12) ∂ θ u ( r e i θ ) (cid:12)(cid:12) g = r (cid:12)(cid:12) ∂ r u (e i θ ) (cid:12)(cid:12) g ≤ π E (2 r ) ∀ r ∈ (0 , δ ) , (4.14) ℓ g ( γ r ) = 128 πE (2 r ) ∀ r ∈ (0 , δ ) . (4.15)By the continuity of u , we can assume that u ( B δ ) ⊂ B gδ ( x ). For r ∈ (0 , δ ) and ρ ∈ (0 , r ), define F ρ ; r : S −→ X to be the map obtained from u | B r − B ρ by attaching disks to the boundary components ∂B r and ∂B ρ and letting F ρ ; r be given by f γ r and f γ ρ on these two disks, respectively; see Figure 7. Since F ρ ; r is homotopic to a constant map and ω is closed,0 = Z S F ∗ ρ ; r ω = E g (cid:0) u ; B r − B ρ (cid:1) + Z B f ∗ γ ρ ω − Z B f ∗ γ r ω . E g (cid:0) u ; B r − B ρ (cid:1) ≤ Cℓ g ( γ r ) + CE (2 ρ ) (4.16)for some C ∈ R + independent of r and ρ as above. Since E g ( u ; B ∗ R ) < E (2 ρ ) −→ ρ −→ ρ −→
0, we thus obtain (4.12).
Corollary 4.10. If ( X, J, g ) , R , and u are as in Proposition 4.8, there exist δ ∈ (0 , R ) and µ, C ∈ R + such that (cid:12)(cid:12) d r e i θ u (cid:12)(cid:12) g ≤ Cr µ − ∀ r ∈ (0 , δ ) . (4.17) Proof.
Let γ r , δ , C , and E ( r ) be as in the statement and proof of Lemma 4.9. Thus, E ( r ) ≡ Z π Z r (cid:12)(cid:12) d ρ e i θ u (cid:12)(cid:12) g ρ d ρ d θ ≤ Cℓ g ( γ r ) = 12 Cr (cid:18) Z π (cid:12)(cid:12) d r e i θ u (cid:12)(cid:12) g d θ (cid:19) ≤ Cπr Z π (cid:12)(cid:12) d r e i θ u (cid:12)(cid:12) g d θ = 2 CπrE ′ ( r ) ∀ r ∈ (0 , δ ) . This implies that (cid:0) r − / Cπ E ( r ) (cid:1) ′ ≥ , E ( r ) ≤ δ − / Cπ E ( δ ) · r / Cπ ≡ C ′ r µ ∀ r ∈ (0 , δ ) . Combining this with (4.14), we obtain (4.17) with δ replaced by δ/ Proof of Proposition 4.8 . With µ as in Corollary 4.10, let p ∈ R + be such that p > − µ ) p <
2. In particular, u | B R/ ∈ L p (cid:0) B R/ ; X (cid:1) , ¯ ∂ J u | B R/ = 0 ∈ L p (cid:0) B R/ ; X (cid:1) . By elliptic regularity, this implies that u is smooth; see [12, Theorem B.4.1]. By the continuity of¯ ∂ J u , u is then J -holomorphic on all of B R . J -holomorphic maps We next combine the local statement of Proposition 3.1 and some of its implications with theregularity statement of Proposition 4.8 to obtain a global description of J -holomorphic maps. Proposition 4.11.
Let ( X, J ) be an almost complex manifold, (Σ , j ) be a compact Riemann surface, u : Σ −→ X be a J -holomorphic map. If u is simple, then u is somewhere injective and all limitpoints of the set (cid:8) z ∈ Σ : | u − ( u ( z )) | > (cid:9) (4.18) are critical points of u . Suppose (
X, J ) is an almost complex manifold, (Σ , j ) is a Riemann surface, and u : Σ −→ X is a J -holomorphic map. Let Σ ∗ u = Σ − u − (cid:0) u (cid:0) { z ∈ Σ : d z u = 0 } (cid:1)(cid:1) (4.19)be the preimage of the regular values of u and R ∗ u ⊂ Σ ∗ u × Σ ∗ u
30e the subset of pairs ( z, z ′ ) such that there exists a diffeomorphism ϕ z ′ z : U z −→ U z ′ betweenneighborhoods of z and z ′ in Σ satisfying ϕ z ′ z ( z ) = z ′ and u | U z = u ◦ ϕ z ′ z . (4.20)Denote by R u ⊂ Σ × Σ the closure of R ∗ u .It is immediate that R ∗ u is an equivalence relation on Σ and u ( z ) = u ( z ′ ) whenever ( z, z ′ ) ∈ R ∗ u .Thus, R u is also a reflexive and symmetric relation and u ( z ) = u ( z ′ ) whenever ( z, z ′ ) ∈ R u . ByLemma 4.14 below, R u is transitive as well. We denote this equivalence relation by ∼ u . Let h u : Σ −→ Σ ′ ≡ Σ / ∼ u and u ′ : Σ ′ −→ X (4.21)be the quotient map and the continuous map induced by u , respectively. In particular, u = u ′ ◦ h u : Σ −→ X. In the case Σ is compact, we will show that Σ ′ inherits a Riemann surface structure j ′ from j sothat the maps h u and u ′ are j ′ - and J -holomorphic, respectively. If the degree of h is 1, we willshow that all limit points of the set (4.18) are critical points of u . Lemma 4.12.
Suppose ( X, J ) is an almost complex manifold, R ∈ R + , and u : B R −→ X is anon-constant J -holomorphic map such that d z u = 0 for all z ∈ B ∗ R . Then there exist m ∈ Z + and aneighborhood U of in B R such that h u : U ∩ B ∗ R −→ h u (cid:0) U ∩ B ∗ R (cid:1) ⊂ B ′ R (4.22) is a covering projection of degree m .Proof. By the continuity of u , we can assume that X = C n , u (0) = 0, and J = J C n . As shown inthe proof of Corollary 3.11, there exist ǫ ∈ (0 , R ) and δ ∈ (0 , ǫ/
2) such that U ≡ u − (cid:0) u ( B δ ) (cid:1) ∩ B ǫ ⊂ B δ . By Proposition 3.1 and the compactness of B δ ⊂ B R , the number m ( z ) ≡ (cid:12)(cid:12) h − u ( h u ( z )) ∩ U (cid:12)(cid:12) is finite for every z ∈ U ∩ B ∗ R .Suppose z i ∈ B ∗ δ and z ′ i ∈ U are sequences such that z i converges to some z ∈ B ∗ δ with z i = z forall i and h u ( z i ) = h u ( z ′ i ) for all i . Passing to a subsequence, we can assume that z ′ i converges tosome z ′ ∈ B δ . By the continuity of u , u ( z ′ ) = u ( z ) and so z ′ ∈ U . Corollary 3.10 then impliesthat h u ( z ′ ) = h u ( z ). Since B ∗ δ is connected, this implies that the number m ( z ) is independent of z ∈ U ∩ B ∗ R ; we denote it by m .Suppose z ∈ U ∩ B ∗ R and h − u (cid:0) h u ( z ) (cid:1) ∩ U = (cid:8) z , . . . , z m (cid:9) . Let ϕ i : U −→ U i for i = 1 , . . . , m be diffeomorphisms between neighborhoods of z and z i in U ∩ B ∗ R such that ϕ i ( z ) = z i , u = u ◦ ϕ i ∀ i, U i ∩ U j = ∅ ∀ i = j, u : U −→ X is injective. Then h u ( U ) ⊂ B ′ R is an open neighborhood of h u ( z ), h − u (cid:0) h u ( U ) (cid:1) ∩ U = m G i =1 U i , and h u : U i −→ h u ( U ) is a homeomorphism. Thus, (4.22) is a covering projection of degree m . Lemma 4.13.
Suppose ( X, J ) , R , and u are as in Lemma 4.12. Then there exists a neighbor-hood U of in B R such that Ψ : h u ( U ) −→ C , h u ( z ) = Y z ′ ∈ h − u ( h u ( z )) ∩ U z ′ , (4.23) is a homeomorphism from an open neighborhood of h u (0) in B ′ R to an open neighborhood of 0 in C and Ψ ◦ h u | U is a holomorphic map.Proof. By Lemma 4.12, there exists a neighborhood U of 0 in B R so that (4.22) is a coveringprojection of some degree m ∈ Z + . Since the restriction of u to B ∗ R is a J -holomorphic immersion,the diffeomorphisms ϕ i as in the proof of Lemma 4.12 are holomorphic. Thus, the mapΨ ◦ h u | U ∩ B ∗ R : U ∩ B ∗ R −→ C , z −→ Y z ′ ∈ h − u ( h u ( z )) ∩ U z ′ is holomorphic. Since it is also bounded, it extends to a holomorphic map e Ψ : U −→ C . This extension is non-constant and vanishes at 0.After possibly shrinking U , we can assume that there exist k ∈ Z + and C ∈ R + such that C − k | z | k ≤ (cid:12)(cid:12) e Ψ ( z ) (cid:12)(cid:12) ≤ C k | z | k ∀ z ∈ U . (4.24)Since e Ψ ( z ′ ) = e Ψ ( z ) for all z ′ ∈ h − u ( h u ( z )) ∩ U , it follows that C − | z | ≤ | z ′ | ≤ C | z | ∀ z ′ ∈ h − u ( h u ( z )) ∩ U , z ∈ U ,C − m | z | m ≤ (cid:12)(cid:12) e Ψ ( z ) (cid:12)(cid:12) ≤ C m | z | m ∀ z ∈ U . Along with (4.24), the last estimate implies that k = m and that e Φ has a zero of order precisely m at z = 0. Thus, shrinking δ in the proof of Lemma 4.12 if necessary, we can assume that e Φ is m : 1 over U ∩ B ∗ R . This implies that the map (4.23) and its extension over the closure of h u ( U )in B ′ R are continuous and injective. Since the closure of h u ( U ) is compact and C is Hausdorff, weconclude that (4.23) is a homeomorphism onto an open subset of C . Lemma 4.14.
Suppose ( X, J ) , (Σ , j ) , and u are as in Proposition 4.11 and ( x, y ) ∈ R u . For everyneighborhood U x of x in Σ , the image of the projection R u ∩ ( U x × X ) −→ X to the second component contains a neighborhood U y of y in Σ . roof. By Corollary 3.4, the last set in (4.19) is finite. By the same reasoning as in the last partof the proof of Lemma 4.12, h u : Σ ∗ u −→ h u (Σ ∗ u ) ⊂ Σ ′ (4.25)is a local homeomorphism. Since u ( z ) = u ( z ′ ) for all ( z, z ′ ) ∈ R ∗ u , the definition of Σ ∗ u thus impliesthat (4.25) is a finite-degree covering projection over each topological component of h u (Σ ∗ u ). Sincethe complement of finitely many points in a connected Riemann surface is connected, the degreeof this covering over a point h u ( z ) depends only on the topological component of Σ containing z .For any point z ∈ Σ, not necessarily in Σ ∗ u , we denote this degree by d ( z ).By Corollary 3.4, the set S ≡ u − (cid:0) u ( x ) (cid:1) ⊂ Σis finite. Let W ⊂ X be a neighborhood of u ( x ) such that the topological components Σ s of u − ( W )containing the points s ∈ S are pairwise disjoint (if U is a union of disjoint balls around the pointsof S , then W ≡ X − u (Σ − U )works). By Lemma 4.12, for each s ∈ S there exists a neighborhood U ′ s of s in Σ s such that h u : U ′ s −{ s } −→ h u (cid:0) U ′ s −{ s } (cid:1) ⊂ Σ ′ is a covering projection of some degree m s ∈ Z + ; we can assume that U ′ x ⊂ U x . Along with thecompactness of Σ, the former implies that (cid:12)(cid:12) h − u (cid:0) h u ( y ′ ) (cid:1) ∩ U ′ s (cid:12)(cid:12) ∈ (cid:8) , m s (cid:9) ∀ y ′ ∈ U ′ s ′ ∩ Σ ∗ u , s, s ′ ∈ S, X s ∈ S (cid:12)(cid:12) h − u (cid:0) h u ( y ′ ) (cid:1) ∩ U ′ s (cid:12)(cid:12) = d ( s ′ ) ∀ y ′ ∈ U ′ s ′ ∩ Σ ∗ u , s ′ ∈ S. (4.26)Define P y ( S ) = (cid:8) S ′ ⊂ S : X s ∈ S ′ m s = d ( y ) (cid:9) . Let U ′′ y ⊂ U ′ y be a connected neighborhood of y . For each S ′ ∈ P y ( S ), define U ′′ y ; S ′ = (cid:8) y ′ ∈ U ′′ y ∩ Σ ∗ u : { s ∈ S : h − u ( h u ( y ′ )) ∩ U ′ s = ∅} = S ′ (cid:9) . By (4.26), these sets partition U ′′ y ∩ Σ ∗ u . Since each set (cid:8) y ′ ∈ U ′′ y ∩ Σ ∗ u : h − u ( h u ( y ′ )) ∩ U ′ s = ∅ (cid:9) is open, (4.26) also implies that each set U ′′ y ; S ′ is open. Since the set U ′′ y ∩ Σ ∗ u is connected, it followsthat U ′′ y ∩ Σ ∗ u = U ′′ y ; S y for some S y ∈ P y ( S ). Since ( x, y ) ∈ R u , x ∈ S y . Thus, the image of theprojection R u ∩ ( U ′ x × X ) −→ X to the second component contains the neighborhood U ′′ y of y in Σ. Corollary 4.15.
Suppose ( X, J ) , (Σ , j ) , and u are as in Proposition 4.11. The quotient map h u in (4.21) is open and closed. roof. The openness of h u is immediate from Lemma 4.14. Suppose A ⊂ Σ is a closed subset and y i ∈ h − u ( h u ( A )) is a sequence converging to some y ∈ Σ. Let x i ∈ A be such that h u ( x i ) = h u ( y i ).Passing to a subsequence, we can assume that the sequence x i converges to some x ∈ A . SinceΣ − Σ ∗ u consists of isolated points, we can also assume that y i ∈ Σ ∗ u and so ( x i , y i ) ∈ R ∗ u . Thus,( x, y ) ∈ R u and so y ∈ h − u ( h u ( A )). We conclude that h u is a closed map. Proof of Proposition 4.11 . Let Σ ′ , h u , and u ′ be as in (4.21). By the second statement inCorollary 4.15 and [13, Lemma 73.3], Σ ′ is a Hausdorff topological space. Fix a Riemannian met-ric g on X .For ( z, z ′ ) ∈ R ∗ u with z = z ′ , the neighborhoods U z and U z ′ as in (4.20) can be chosen so that theyare disjoint and u | U z is an embedding. Since u is J -holomorphic, ϕ z ′ z is then a biholomorphicmap and h u | U z is a homeomorphism onto h u ( U z ) ⊂ Σ ′ . Thus, the Riemann surface structure j on Σ determines a Riemann surface structure j ′ on h u (Σ ∗ u ) so that h u | Σ ∗ u is a holomorphic coveringprojection of h u (Σ ∗ u ) and u ′ | h u (Σ ∗ u ) is a J -holomorphic map with E g (cid:0) u ′ ; h u (Σ ∗ u ) (cid:1) ≤ E g ( u ) . (4.27)By Corollary 3.4, Σ ′ u − h u (Σ ∗ u ) consists of finitely many points. By the first statement in Corol-lary 4.15 and by Lemma 4.13, j ′ extends over these points to a Riemann surface structure on Σ ′ ; wedenote the extension also by j ′ . Since the continuous map h u is j ′ -holomorphic outside of the finitelymany points of Σ − Σ ∗ u , it is holomorphic everywhere. Since the continuous map u ′ is J -holomorphicon h u (Σ ∗ u ), (4.27) and Proposition 4.8 imply that it is J -holomorphic everywhere.Suppose z ∈ Σ and z i , z ′ i ∈ Σ with i ∈ Z + are such thatd z u = 0 , z i = z ′ i , u ( z i ) = u ( z ′ i ) ∀ i, lim i −→∞ z i = z. Passing to a subsequence, we can assume that the sequence z ′ i converges to some point z ′ ∈ Σwith u ( z ′ ) = u ( z ). Since the restriction of u to a neighborhood of z is an embedding, z ′ = z . ByCorollary 3.10, there exists a diffeomorphism ϕ z ′ z as in (4.20). Thus, h u ( z ) = h u ( z ′ ), the map h u isnot injective, and u is not simple. Proposition 4.16 and Corollary 4.17 below concern J -holomorphic maps from long cylinders. Theirsubstance is that most of the energy and variation of such maps are concentrated near the ends.These technical statements are used to obtain important geometric conclusions in Sections 5.2and 5.3. Proposition 4.16. If ( X, J ) is an almost complex manifold and g is a Riemannian metric on X ,then there exist continuous functions δ J,g , ~ J,g , C
J,g : X −→ R + with the following properties. If u : [ − R, R ] × S −→ X is a J -holomorphic map such that Im u ⊂ B gδ J,g ( u (0 , ( u (0 , , then E g (cid:0) u ; [ − R + T, R − T ] × S (cid:1) ≤ C J,g (cid:0) u (1 , (cid:1) e − T E g ( u ) ∀ T ≥ . (4.28) If in addition E g ( u ) < ~ J,g (cid:0) u (0 , (cid:1) , then diam g (cid:0) u ([ − R + T, R − T ] × S ) (cid:1) ≤ C J,g (cid:0) u (1 , (cid:1) e − T/ q E g ( u ) ∀ T ≥ . (4.29)34 orollary 4.17. If ( X, J ) is a compact almost complex manifold and g is a Riemannian metricon X , there exist ~ J,g , C
J,g ∈ R + with the following property. If u : [ − R, R ] × S −→ X is a J -holomorphic map such that E g ( u ) < ~ J,g , then E g (cid:0) u ; [ − R + T, R − T ] × S (cid:1) ≤ C J,g e − T E g ( u ) ∀ T ≥ , diam g (cid:0) u ([ − R + T, R − T ] × S ) (cid:1) ≤ C J,g e − T/ q E g ( u ) ∀ T ≥ . As an example, the energy of the injective map[ − R, R ] × S −→ C , ( s, θ ) −→ s e i θ , is the area of its image, i.e. π (e R − e − R (cid:1) . Thus, the exponent e − T in (4.28) can be replaced by e − T in this case. The proof of Proposition 4.16 shows that in general the exponent can be taken tobe e − µT with µ arbitrarily close to 2, but at the cost of increasing C J,g and reducing δ J,g . Lemma 4.18 (Poincare Inequality) . If f : S −→ R n is a smooth function such that R π f ( θ )d θ = 0 ,then Z π | f ( θ ) | d θ ≤ Z π | f ′ ( θ ) | d θ. Proof:
We can write f ( θ ) = k< ∞ P k> −∞ a k e i kθ . Since R π f ( θ )d θ = 0, a = 0. Thus, Z π | f ( θ ) | d θ = k< ∞ X k> −∞ | a k | ≤ k< ∞ X k> −∞ | ka k | = Z π | f ′ ( θ ) | d θ. Proof of Proposition 4.16 . It is sufficient to establish the first statement under the assumptionthat (
X, g ) is C n with the standard Riemannian metric, J agrees with the standard complexstructure J C n at 0 ∈ C n , and u (0 ,
1) = 0. Let¯ ∂u = 12 (cid:0) u t + J C n u θ (cid:1) . By our assumptions, there exist δ ′ , C > u (0 , (cid:12)(cid:12) ¯ ∂ z u (cid:12)(cid:12) ≤ Cδ (cid:12)(cid:12) d z u (cid:12)(cid:12) ∀ z ∈ u − (cid:0) B δ (0) (cid:1) , δ ≤ δ ′ . (4.30)Write u = f + i g , with f, g taking values in R n and assume that Im u ⊂ B δ (0). By (2.4), | d u | = 4 (cid:12)(cid:12) ¯ ∂u (cid:12)(cid:12) + 2d( f · d g ) . Combining this with (4.30) and Stokes’ Theorem, we obtain Z [ − t,t ] × S | d u | ≤ C δ Z [ − t,t ] × S | d u | + 2 Z { t }× S f · g θ d θ − Z {− t }× S f · g θ d θ . (4.31)Let e f = f − π R π f d θ . By H¨older’s inequality and Lemma 4.18, (cid:12)(cid:12)(cid:12)(cid:12) Z {± t }× S f · g θ d θ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z {± t }× S e f · g θ d θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) Z {± t }× S | e f | d θ (cid:19) (cid:18) Z {± t }× S | g θ | d θ (cid:19) ≤ (cid:18) Z {± t }× S | e f θ | d θ (cid:19) (cid:18) Z {± t }× S | g θ | d θ (cid:19) ≤ Z {± t }× S | u θ | d θ . (4.32)35ince 3 | u θ | = 2 | u θ | + (cid:12)(cid:12) u t − ∂u (cid:12)(cid:12) ≤ | d u | + 8 (cid:12)(cid:12) ¯ ∂u (cid:12)(cid:12) , the inequalities (4.30)-(4.32) give (cid:0) − C δ (cid:1)Z [ − t,t ] × S | d u | ≤ (cid:0) C δ (cid:1)(cid:18) Z { t }× S | d u | d θ + Z {− t }× S | d u | d θ (cid:19) . Thus, the function ε ( T ) ≡ E g (cid:0) u ; [ − R + T, R − T ] (cid:1) ≡ Z [ − R + T,R − T ] × S | d u | d θ d t satisfies ε ( T ) ≤ − ε ′ ( T ) for all T ∈ [ − R, R ], if δ is sufficiently small (depending on C ). This im-plies (4.28).Let h J,g ( x ) = ( x, δ J,g ( x )), with h J,g ( · , · ) as in Proposition 4.1 and δ J,g ( · ) as provided by the previousparagraph. Suppose u also satisfies the last condition in Proposition 4.16. By Proposition 4.1and (4.28), | d ( t,θ ) u | ≤ q E g ( u ; [ −| t |− , | t | +1] × S ) ≤ q C J,g ( u (0 , (1+ | t |− R ) / q E g ( u )for all t ∈ [ − R +1 , R −
1] and θ ∈ S . Thus, for all t , t ∈ [ − R + T, R − T ] with T ≥ θ , θ ∈ S , d g (cid:0) u ( t , θ ) , u ( t , θ ) (cid:1) ≤ q C J,g ( u (0 , E g ( u ) (cid:18) π e (1+ | t |− R ) / + Z t t e (1+ | t |− R ) / d t (cid:19) ≤ (cid:0) π +12 (cid:1)q C J,g ( u (0 , (1 − T ) / q E g ( u ) . This establishes (4.29).
Lemma 4.19. If ( X, J ) is a compact almost complex manifold and g is a Riemannian metricon X , there exists a continuous function ǫ J,g : R + −→ R + with the following property. If δ ∈ R + and u : ( − R, R ) × S −→ X is a J -holomorphic map with E g ( u ) < ǫ J,g ( δ ) , then diam g (cid:0) u (cid:0) [ − R +1 , R − × S (cid:1)(cid:1) ≤ δ . Proof.
By Proposition 3.12 and the compactness of X , there exists c J,g ∈ R + with the followingproperty. If (Σ , j ) is a compact connected Riemann surface with boundary, u : Σ −→ X is a non-constant J -holomorphic map, x ∈ X , and δ ∈ R + are such that u ( ∂ Σ) ∩ B gδ ( x ) = ∅ , then E g ( u ) ≥ c J,g δ . (4.33)Let ~ J,g > ~ J,g in the statement of Proposition 4.1 on thecompact space X × [0 , diam g ( X )].Suppose u : ( − R, R ) × S −→ X is a J -holomorphic map with E g ( u ) < ~ J,g and δ u ≡ diam g (cid:0) u ([ − R +1 , R − × S ) (cid:1) > q E g ( u ) .
36y the first condition on u , (cid:12)(cid:12) d z u (cid:12)(cid:12) g ≤ π E g ( u ) ∀ z ∈ [ − R +1 , R − × S , diam g (cid:0) u ( r × S ) (cid:1) ≤ q E g ( u ) ∀ r ∈ [ − R +1 , R − . (4.34)Let r − , r , r + ∈ [ − R +1 , R −
1] and θ − , θ , θ + ∈ S be such that r − < r < r + , d g (cid:0) u ( r , θ ) , u ( r ± , θ ± ) (cid:1) ≥ δ u . By (4.34), we can apply (4.33) withΣ = [ r − , r + ] × S , x = u ( r , θ ) , δ = 14 δ u , and u replaced by its restriction to Σ. We conclude that E g ( u ) ≥ c J,g δ u . It follows that the function ǫ J,g : R + −→ R + , ǫ J,g ( δ ) = min (cid:18) ~ J,g , δ , c J,g δ (cid:19) , has the desired property. Proof of Corollary 4.17 . Let δ ∈ R + be the minimum of the function δ J,g in Proposition 4.16 and ε J,g ( · ) be as in Lemma 4.19. Take C J,g to be the maximum of the function C J,g in Proposition 4.16times e and ~ J,g ∈ R + to be smaller than the minimum of the function ~ J,g in Proposition 4.16 andthe number ε J,g ( δ ). J -holomorphic maps This section studies the limiting behavior of sequences of J -holomorphic maps from Riemannsurfaces into a compact almost complex manifold ( X, J ). The compactness of X plays an essentialrole in the statements below, in contrast to nearly all statements in Sections 3 and 4, By Cauchy’s Integral Formula, a bounded holomorphic map u : B ∗ R −→ C n extends over the origin.By Proposition 5.1 below, the same is the case for a J -holomorphic map u : B ∗ R −→ X of boundedenergy if X is compact. Proposition 5.1 (Removal of Singularity) . Let ( X, J ) be a compact almost complex manifold and u : B ∗ R −→ X be a J -holomorphic map. If the energy E g ( u ) of u , with respect to any metric g on X ,is finite, then u extends to a J -holomorphic map e u : B R −→ X . A basic example of a holomorphic function u : C ∗ −→ C that does not extend over the origin 0 ∈ C is z −→ /z . The energy of u | B ∗ R with respect to the standard metric on C is given by E (cid:0) u ; B ∗ R (cid:1) = 12 Z B R | d u | = Z B R | z | = Z π Z R r − d r d θ < ∞ . | d u | were replaced by | d u | − ǫ for any ǫ >
0. Thisobservation illustrates the crucial role played by the energy in the theory of J -holomorphic maps.By Cauchy’s Integral Formula, the conclusion of Proposition 5.1 holds if J is an integrable almostcomplex structure and u ( B ∗ δ ) is contained in a complex coordinate chart for some δ ∈ (0 , R ). Wewill use the Monotonicity Lemma to show that the latter is the case if the energy of u is finite; theintegrability of J turns out to be irrelevant here. Proof of Proposition 5.1 . In light of Proposition 4.8, it is to sufficient to show that u extendscontinuously over the origin. Let c J,g , ~ J,g ∈ R + be as in the proof of Lemma 4.19. We can assumethat R = 1 and u is non-constant. Define v : R − × S −→ X, v (cid:0) r, e i θ (cid:1) = u (cid:0) e r + i θ ) . This map is J -holomorphic and satisfies E g ( v ) = E g ( u ) < ∞ .Since E g ( u ) < ∞ , lim r −→−∞ E g (cid:0) v ; ( −∞ , r ) × S (cid:1) = lim r −→−∞ E g (cid:0) u ; B ∗ e r (cid:1) = 0 . (5.1)In particular, there exists R ∈ R − such that E g (cid:0) v ; ( −∞ , r ) × S (cid:1) < ~ J,g ∀ r < R. By Proposition 4.1 and our choice of ~ J,g , this implies that (cid:12)(cid:12) d z v (cid:12)(cid:12) g ≤ π E g (cid:0) v ; ( −∞ , r +1) × S (cid:1) ∀ z ∈ ( −∞ , r ) × S , r < R − , diam g (cid:0) v ( { r }× S ) (cid:1) ≤ √ π q E g ( v ; ( −∞ , r +1) × S ) ∀ r < R − . Combining the last bound with (5.1), we obtainlim r −→−∞ diam g (cid:0) v ( { r }× S ) (cid:1) = 0 . Thus, it remains to show that lim r −→−∞ v ( r,
1) exists.Since X is compact, every sequence in X has a convergent subsequence. Suppose there exist δ ∈ R + , x, y ∈ X, i k , r k ∈ R − s.t. d g ( x, y ) > δ, r k +1 < i k < r k , v (cid:0) { i k }× S (cid:1) ⊂ B δ ( x ) , v (cid:0) { r k }× S (cid:1) ⊂ B δ ( y ) . We thus can apply (4.33) with Σ, x , and u replaced byΣ k ≡ [ r k +1 , r k ] × S , x k ≡ u ( i k , , and v k ≡ v | Σ k , respectively. We conclude that E g ( v ) ≥ X k E g (cid:0) v ; Σ k (cid:1) = X k E g ( v k ) ≥ X k c J,g δ = ∞ . However, this contradicts the assumption that E g ( v ) = E g ( u ) < ∞ .38 r k r k +1 i k B δ ( y ) B δ ( y ) B δ ( x )Σ k vv v Figure 8: Setup for the proof of Proposition 5.1
The next three statements are used in Section 5.2 to show that no energy is lost under Gromov’sconvergence procedure, the resulting bubbles connect, and their number is finite.
Lemma 5.2.
Suppose ( X, J ) is an almost complex manifold with a Riemannian metric g and u i : B −→ X is a sequence of J -holomorphic maps converging uniformly in the C ∞ -topology oncompact subsets of B ∗ to a J -holomorphic map u : B −→ X such that the limit m ≡ lim δ −→ lim i −→∞ E g ( u i ; B δ ) (5.2) exists and is nonzero.(1) The limit m ( δ ) ≡ lim i −→∞ E g ( u i ; B δ ) exists and is a continuous, non-decreasing function of δ .(2) For every sequence z i ∈ B converging to 0, lim i −→∞ E g ( u i ; B δ ( z i )) = m ( δ ) .(3) For every sequence z i ∈ B converging to 0, µ ∈ (0 , m ) , and i ∈ Z + sufficiently large, there existsa unique δ i ( µ ) ∈ (0 , −| z i | ) such that E g ( u i ; B δ i ( µ ) ( z i )) = µ . Furthermore, lim R −→∞ lim δ −→ lim i −→∞ E g (cid:0) u i ; B Rδ ( z i ) − B δ i ( µ ) ( z i ) (cid:1) = m − µ. (5.3) Proof. (1) Since d u i converges uniformly to d u on compact subsets of B ∗ , m ( δ ) ≡ lim i −→∞ E g (cid:0) u i ; B δ (cid:1) = lim δ ′ −→ lim i −→∞ E g (cid:0) u i ; B δ ′ (cid:1) + lim δ ′ −→ lim i −→∞ E g (cid:0) u i ; B δ − B δ ′ (cid:1) = m + lim δ ′ −→ E g (cid:0) u ; B δ − B δ ′ (cid:1) = m + E g ( u ; B δ ) . Since E g ( u ; B δ ) is a continuous, non-decreasing function of δ , so is m ( δ ).(2) For all δ, δ ′ ∈ R + and z i ∈ B δ ′ , B δ − δ ′ ⊂ B δ ( z i ) ⊂ B δ + δ ′ . Thus, E g (cid:0) u i ; B δ − δ ′ (cid:1) ≤ E g (cid:0) u i ; B δ ( z i ) (cid:1) ≤ E g (cid:0) u i ; B δ + δ ′ (cid:1) for all i ∈ Z + sufficiently large andlim δ ′ −→ m ( δ − δ ′ ) ≤ lim δ ′ −→ lim i −→∞ E g (cid:0) u i ; B δ ( z i ) (cid:1) ≤ lim δ ′ −→ m ( δ + δ ′ ) ∀ δ ′ ∈ R + . The claim now follows from (1). 393) By (2), (1), and (5.2), lim i −→∞ E g (cid:0) u i ; B δ ( z i ) (cid:1) = m ( δ ) ≥ m . Thus, there exists i ( µ ) ∈ Z + such that E g ( u i ; B δ ( z i )) > µ ∀ i ≥ i ( µ ) . Since E g ( u i ; B δ ( z i )) is a continuous, increasing function of δ which vanishes at δ = 0, for every i ≥ i ( µ ) there exists a unique δ i ( µ ) ∈ (0 , δ ) such that E g ( u i ; B δ i ( µ ) ( z i )) = µ .By (2), (1), and (5.2),lim R −→∞ lim δ −→ lim i −→∞ E g (cid:0) u i ; B Rδ ( z i ) (cid:1) = lim R −→∞ lim δ −→ m ( Rδ ) = lim R −→∞ m = m . Combining this with the definition of δ i ( µ ), we obtain (5.3). Corollary 5.3. If ( X, J ) is a compact almost complex manifold with a Riemannian metric g , thenthere exists ~ J,g ∈ R + with the following properties. If u i : B −→ X is a sequence of J -holomorphicmaps converging uniformly in the C ∞ -topology on compact subsets of B ∗ to a J -holomorphic map u : B −→ X such that lim i −→∞ max B / (cid:12)(cid:12) d u i (cid:12)(cid:12) g = ∞ and the limit (5.2) exists, then(1) m ≥ ~ J,g ;(2) for every sequence z i ∈ B δ converging to 0 and µ ∈ ( m − ~ J,g , m ) , the numbers δ i ( µ ) ∈ (0 , −| z i | ) of Lemma 5.2(3) satisfy lim R −→∞ lim i −→∞ E g ( u i ; B Rδ i ( µ ) ( z i )) = m , (5.4)lim R −→∞ lim δ −→ lim i −→∞ diam g (cid:0) u i ( B δ ( z i ) − B Rδ i ( µ ) ( z i )) (cid:1) = 0 . (5.5) Proof.
Let ~ J,g be the smaller of the constants ~ J,g in Corollaries 4.2 and 4.17. Let u i , u , and m be as in the statement of the corollary.(1) For each i ∈ Z + , let M i = max B / (cid:12)(cid:12) d z u i (cid:12)(cid:12) g ∈ R + and z i ∈ B / be such that | d z i u i | g = M i . Since M i −→ ∞ as i −→ ∞ and u i converges uniformly inthe C ∞ -topology on compact subsets of B ∗ to u , z i −→
0. For i ∈ Z + such that | z i | +1 / √ M i < / v i : B √ M i −→ X, v i ( z ) = u i (cid:0) z i + z/M i (cid:1) . Thus, v i is a J -holomorphic map withsup (cid:12)(cid:12) d v i (cid:12)(cid:12) g = (cid:12)(cid:12) d v i (cid:12)(cid:12) g = 1 , E g ( v i ) = E g (cid:0) u i ; B / √ M i ( z i ) (cid:1) ≤ E g (cid:0) u i ; B | z i | +1 / √ M i (cid:1) . (5.6)40y the first statement in (5.6) and the ellipticity of the ¯ ∂ -operator, a subsequence of v i convergesuniformly in the C ∞ -topology on compact subsets of C to a non-constant J -holomorphic map v : C −→ X . By the second statement in (5.6) and Lemma 5.2(1), E g ( v ) ≤ lim sup i −→∞ E g (cid:0) u i ; B / √ M i ( z i ) (cid:1) ≤ lim δ −→ lim i −→∞ E g (cid:0) u i ; B δ (cid:1) = m . (5.7)By Proposition 5.1, v thus extends to a J -holomorphic map e v : P −→ X . By Corollary 4.2, E g ( v ) = E g ( e v ) ≥ ~ J,g . Combining this with (5.7), we obtain the first claim.(2) By the first two statements in Lemma 5.2 and (5.2),lim δ −→ lim i −→∞ E g (cid:0) u i ; B δ ( z i ) (cid:1) = lim δ −→ m ( δ ) = m . (5.8)After passing to a subsequence of u i , we can thus assume that there exists a sequence δ i −→ i −→∞ E g ( u i ; B δ i ( z i )) = m . (5.9)Since δ i −→
0, (5.8) and (5.9) imply thatlim R −→∞ lim i −→∞ E g ( u i ; B Rδ i ( z i )) = m . (5.10)Suppose µ ∈ ( m − ~ J,g , m ). By (5.10) and the definition of δ i ( µ ),lim R −→∞ lim i −→∞ E g (cid:0) u ; B Rδ i ( z i ) − B δ i ( µ ) ( z i ) (cid:1) = m − µ < ~ J,g . Thus, Corollary 4.17 applies with (
R, T ) replaced by ( ln( Rδ i /δ i ( µ )) , ln R ) and u replaced by the J -holomorphic map v : ( − R, R ) × S −→ X, v (cid:0) r, e i θ (cid:1) = u (cid:0) z i + p Rδ i δ i ( µ ) e r + i θ (cid:1) . By the first statement of Corollary 4.17, E g (cid:0) u ; B δ i ( z i ) (cid:1) − E g (cid:0) u ; B Rδ i ( µ ) ( z i ) (cid:1) = E g (cid:0) u ; B δ i ( z i ) − B Rδ i ( µ ) ( z i ) (cid:1) ≤ C J,g
R E g ( u )for all i sufficiently large (depending on R ); see Figure 9. Combining this with (5.9), we obtain (5.4).It remains to establish (5.5). By (5.3), for all R > δ > R )there exists i ( R, δ ) ∈ Z + such that E g (cid:0) u i ; B Rδ ( z i ) − B δ i ( µ ) ( z i ) (cid:1) < ~ J,g ∀ i > i ( R, δ ) . Thus, Corollary 4.17 applies with (
R, T ) replaced by ( ln( Rδ/δ i ( µ )) , ln R ) and u replaced by the J -holomorphic map v : ( − R, R ) × S −→ X, v (cid:0) r, e i θ (cid:1) = u (cid:0) z i + p Rδδ i ( µ ) e r + i θ (cid:1) . By the second statement of Corollary 4.17,diam g (cid:0) u i ( B δ ( z i ) − B Rδ i ( µ ) ( z i )) (cid:1) ≤ C J,g √ R ~ J,g ∀ i > i ( R, δ ) . This gives (5.5). 41 ln δ i +ln R m ln δ i µ ln δ i ( µ ) µ ∗ δ i ( µ )+ln R disappearingenergy Figure 9: Illustration for the proof of (5.4)
Lemma 5.4. If ( X, J ) is a compact almost complex manifold with a Riemannian metric g , thenthere exists a function N : R −→ Z with the following property. If (Σ , j ) is compact Riemann surface, S ⊂ Σ is a finite subset, and u i : U i −→ X is a sequence of J -holomorphic maps from open subsetsof Σ with U i ⊂ U i +1 , Σ − S = ∞ [ i =1 U i , and E ≡ lim inf i −→∞ E g ( u i ) < ∞ , (5.11) then there exist a subset S ⊂ Σ with | S | ≤ N ( E )+ | S | and a subsequence of u i converging uniformlyin the C ∞ -topology on compact subsets of Σ − S to a J -holomorphic map u : Σ −→ X .Proof. Let ~ J,g be the minimal value of the function provided by Proposition 4.1. For E ∈ R + , let N ( E ) ∈ Z ≥ be the smallest integer such that E ≤ N ( E ) ~ J,g .Let Σ, S , u i , and E be as in the statement of the lemma and N = N ( E )+ | S | . Fix a Riemannianmetric g Σ on Σ. For z ∈ Σ and δ ∈ Σ, let B δ ( z ) ⊂ Σ denote the ball of radius δ around z . ByProposition 4.1, there exists C ∈ R + with the following property. If u : Σ −→ X is a J -holomorphicmap, z ∈ Σ, and δ ∈ R + , then E g (cid:0) u ; B δ ( z ) (cid:1) < ~ J,g = ⇒ (cid:12)(cid:12) d z u (cid:12)(cid:12) g ≤ C/δ . (5.12)For every pair i, j ∈ Z + , let { z kij } Nk =1 be a subset of points of Σ containing S such that z ∈ Σ ∗ ij ≡ Σ − N [ k =1 B /j (cid:0) z kij (cid:1) = ⇒ E g (cid:0) u i ; B /j ( z ) ∩ U i (cid:1) < ~ J,g . (5.13)By (5.12) and (5.13), (cid:12)(cid:12) d z u i (cid:12)(cid:12) g ≤ Cj ∀ z ∈ Σ ∗ ij s.t. B /j ( z ) ⊂ U i . (5.14)After passing to a subsequence of { u i } , we can assume that the sequence E g ( u i ) converges to E and that the sequence { z kij } i ∈ Z + converges to some z kj ∈ Σ for every k = 1 , . . . , N and j ∈ Z + . Alongwith (5.14) and the first two assumptions in (5.11), this implies thatlim sup i −→∞ (cid:12)(cid:12) d z u i (cid:12)(cid:12) g ≤ Cj ∀ z ∈ Σ ∗ ij . (5.15)After passing to another subsequence of { u i } , we can assume that the sequence { z kj } j ∈ Z + convergesto some z k ∈ Σ for every k = 1 , . . . , N . 42y (5.15) and the ellipticity of the ¯ ∂ -operator, a subsequence of u i converges uniformly in the C ∞ -topology on compact subsets of Σ ∗ to a J -holomorphic map v : Σ ∗ −→ X . By (5.15) and theellipticity of the ¯ ∂ -operator, a subsequence of this subsequence in turn converges uniformly in the C ∞ -topology on compact subsets of Σ ∗ to a J -holomorphic map v : Σ ∗ −→ X . Continuing in thisway, we obtain a subsequence of u i converging uniformly in the C ∞ -topology on compact subsetsof Σ ∗ j to a J -holomorphic map v j : Σ ∗ j −→ X for every j ∈ Z + . The limiting maps satisfy v j | Σ j ∩ Σ ∗ j ′ = v j ′ | Σ ∗ j ∩ Σ ∗ j ′ ∀ j, j ′ ∈ Z + . Thus, the map u : Σ ∗ ≡ Σ ∗ − (cid:8) z k (cid:9) −→ X, u ( z ) = v j ( z ) ∀ z ∈ Σ ∗ j , is well-defined and J -holomorphic.By construction, the final subsequence of u i converges uniformly in the C ∞ -topology on compactsubsets of Σ ∗ to u . This implies that E g ( u ) ≤ lim inf i −→∞ E g ( u i ) = E .
By Proposition 5.1, u thus extends to a J -holomorphic map Σ −→ X . We next show that a sequence of maps as in Corollary 5.3 gives rise to a continuous map froma tree of spheres attached at 0 ∈ B , i.e. a connected union of spheres that have a distinguished,base component and no loops; the distinguished component will be attached at ∞ ∈ S to 0 ∈ B .The combinatorial structure of such a tree is described by a finite rooted linearly ordered set , i.e. apartially ordered set ( I, ≺ ) such that(RS1) there is a minimal element ( root ) i ∈ I , i.e. i ≺ h for every h ∈ I −{ i } , and(RS2) for all h , h , i ∈ I with h , h ≺ i , either h = h , or h ≺ h , or h ≺ h .For each i ∈ I − { i } , let p ( i ) ∈ I denote the immediate predecessor of i , i.e. p ( i ) ∈ I such that h ≺ p ( i ) ≺ i for all h ∈ I − { p ( i ) } such that h ≺ i . Such p ( i ) ∈ I exists by (RS1) and is uniqueby (RS2). In the first diagram in Figure 10, the vertices (dots) represent the elements of a rootedlinearly ordered set ( I, ≺ ) and the edges run from i ∈ I −{ i } down to p ( i ).Given a finite rooted linearly ordered set ( I, ≺ ) with minimal element i and a function z : I −{ i } −→ C , i −→ z i , s.t. (cid:0) p ( i ) , z i (cid:1) = (cid:0) p ( i ) , z i (cid:1) ∀ i , i ∈ I −{ i } , i = i , (5.16)let Σ = (cid:18) G i ∈ I { i }× S (cid:19). ∼ , ( i, ∞ ) ∼ (cid:0) p ( i ) , z i ) ∀ i ∈ I −{ i } ;see the second diagram in Figure 10. Thus, the tree of spheres Σ is obtained by attaching ∞ inthe sphere indexed by i to z i in the sphere indexed by p ( i ). The last condition in (5.16) insuresthat Σ is a nodal Riemann surface, i.e. each non-smooth point ( node ) has only two local branches(pieces homeomorphic to C ). 43 p ( i ) i ∞ i ∞ p ( i ) z i Figure 10: A rooted linearly ordered set and an associated tree of spheres
Proposition 5.5.
Let ( X, J ) be a compact almost complex manifold with a Riemannian metric g and u i : B −→ X be a sequence of J -holomorphic maps converging uniformly in the C ∞ -topologyon compact subsets of B ∗ to a J -holomorphic map u : B −→ X . If the limit m ≡ lim δ −→ lim i −→∞ E g ( u i ; B δ ) (5.17) exists and is nonzero, then there exist(1) a nodal Riemann surface (Σ ∞ , j ∞ ) consisting of B with a tree of Riemann spheres P attachedat ∈ B ,(2) a J -holomorphic map u ∞ : Σ ∞ −→ X ,(3) a subsequence of { u i } still denoted by { u i } , and(4) a biholomorphic map ψ i : U i −→ B / , where U i ⊂ C is an open subset,such that(4a) E g ( u ∞ ; Σ ∞ − B ) = m , U i ⊂ U i +1 , and C = S ∞ i =1 U i ,(4b) u i ◦ ψ i converges to u ∞ uniformly in the C ∞ -topology on compact subsets of the complementof the nodes ∞ , w ∗ , . . . , w ∗ k in the sphere P attached at ∈ B ,(4c) if u ∞ | P is constant, P contains at least three nodes of Σ ∞ ;(4d) (4) applies with B , ( { u i } , , and m replaced by a neighborhood of w ∗ r in C , ( { u i ◦ ψ i } , w ∗ r ) ,and m ′ r ≡ lim δ −→ lim i −→∞ E g (cid:0) u i ◦ ψ i ; B δ ( w ∗ r ) (cid:1) , (5.18) respectively, for each r = 1 , . . . , k .Proof. Let ~ J,g be the smaller of the numbers ~ J,g in Corollaries 4.2 and 5.3. In particular, m ≥ ~ J,g .For each i ∈ Z + sufficiently large, choose z i ∈ B / so thatmax z ∈ B / (cid:12)(cid:12) d u i (cid:12)(cid:12) g = (cid:12)(cid:12) d z i u i (cid:12)(cid:12) g . (5.19)44ince u i converges uniformly in the C ∞ -topology on compact subsets of B ∗ to u , z i −→ i −→ ∞ .Thus, B / ( z i ) ⊂ B for all i ∈ Z + sufficiently large. By Lemma 5.2(3), for all i ∈ Z + sufficientlylarge there exists δ i ∈ (0 , /
2) such that E g (cid:0) u i ; B δ i ( z i ) (cid:1) = m − ~ J,g . (5.20)Define ψ i : U i ≡ (cid:8) w ∈ C : z i + δ i w ∈ B / (cid:9) −→ B / by ψ i ( w ) = z i + δ i w . Since δ i −→
0, the second property in (4a) holds.For each i ∈ Z + sufficiently large, let v i = u i ◦ ψ i : U i −→ X. Since u i is J -holomorphic and ψ i is biholomorphic onto its image, v i is a J -holomorphic map with E g ( v i ) = E g ( u i ; B / ). Along with Lemma 5.2(2), this implies thatlim i −→∞ E g ( v i ) = m (1 / < ∞ . By Lemma 5.4, there thus exist a finite collection w ∗ , . . . , w ∗ k ∈ C of distinct points and a subsequenceof { u i } , still denoted by { u i } , such that v i converges uniformly in the C ∞ -topology on compactsubsets of P −{∞ , w ∗ , . . . , w ∗ ℓ } to a J -holomorphic map u : P −→ X . In particular, (4b) holds and | d v i | g is uniformly bounded on compact subsets of P −{∞ , w ∗ , . . . , w ∗ ℓ } . We can also assume thatthe limit (5.18) exists for every r = 1 , . . . , k . We note that E g ( v ) + k X r =1 m ′ r = lim R −→∞ lim δ −→ lim i −→∞ E g (cid:0) v i , B R − k [ r =1 B δ ( w ∗ r ) (cid:1) + k X r =1 lim δ −→ lim i −→∞ E g (cid:0) v i ; B δ ( w ∗ r ) (cid:1) = lim R −→∞ lim i −→∞ E g (cid:0) v i , B R (cid:1) = lim R −→∞ lim i −→∞ E g (cid:0) u i , B Rδ i ( z i ) (cid:1) = m ; (5.21)the last equality holds by (5.4).Let δ ∈ R + be such that the balls B δ ( w ∗ r ) are pairwise disjoint. Iflim sup i −→∞ max B δ ( w ∗ r ) (cid:12)(cid:12) d v i (cid:12)(cid:12) < ∞ for some r , then { v i } converges uniformly in the C ∞ -topology on B δ ( w ∗ r ) to v by the ellipticity ofthe ¯ ∂ -operator. Thus, we can assume thatlim i −→∞ sup B δ ( w ∗ r ) (cid:12)(cid:12) d v i (cid:12)(cid:12) = ∞ for every r = 1 , . . . , k . In light of Corollary 5.3(1), m ′ r ≥ ~ J,g .We next show that u (0) = v ( ∞ ), i.e. that the bubble ( P , v ) connects to ( B , u ) at z = 0. Note that d g (cid:0) u (0) , v ( ∞ ) (cid:1) = lim R −→∞ lim δ −→ d g (cid:0) u ( δ ) , v ( R ) (cid:1) = lim R −→∞ lim δ −→ lim i −→∞ d g (cid:0) u i ( z i + δ ) , v i ( R ) (cid:1) = lim R −→∞ lim δ −→ lim i −→∞ d g (cid:0) u i ( z i + δ ) , u i ( z i + Rδ i ) (cid:1) ≤ lim R −→∞ lim δ −→ lim i −→∞ diam g (cid:0) u i ( B δ ( z i ) − B Rδ i ( z i )) (cid:1) . δ /δ i m − ~ ~ − ǫ i Figure 11: The energy distribution of the rescaled map v i in the proof of Proposition 5.5Along with (5.5), this implies that u (0) = v ( ∞ ).Suppose v : P −→ X is a constant map. By (5.21), k ≥ w ∗ ∈ C such that | d w ∗ v i | −→ ∞ as i −→ ∞ . By (5.19) and the definition of ψ i , | d v i | ≥ | d w v i | for all w ∈ C containedin the domain of v i and so | d v i | −→ ∞ as i −→ ∞ . By (5.18) and (5.20), m ′ ≡ lim δ −→ lim i −→∞ E g ( v i ) ≤ lim i −→∞ E g (cid:0) v i ; B (cid:1) = lim i −→∞ E g (cid:0) u i ; B δ i ( z i ) (cid:1) = m − ~ < m , and so k ≥
2, as claimed in (4c). Since the amount of energy of v i contained in C − B approaches ~ J,g /
2, as illustrated in Figure 11, there must be in particular a bubble point w ∗ r with | w ∗ r | = 1,though this is not material.The above establishes Proposition 5.5 whenever k = 0 by taking u ∞ (cid:12)(cid:12) B = u and u ∞ (cid:12)(cid:12) P = v. Since m ′ r ≥ ~ J,g for every r , k = 0 if m < ~ J,g . If k ≥ m ′ r ≤ m − ~ J,g by (5.21) because E g ( v ) ≥ ~ J,g if v is not constant by Corollary 4.2 and k ≥ m / ~ J,g ] ∈ Z + , we can assume that Proposition 5.5 holds when applied to { v i } on B δ ( w ∗ r ) ⊂ C with r = 1 , . . . , k . This yields a tree Σ r of Riemann spheres P with a distinguished smooth point ∞ and a J -holomorphic map v r : Σ r −→ X such v r ( ∞ ) = v ( w ∗ r ) and E g ( v r ) = m ′ r . Combining the lastequality with (5.21), we obtain E g ( v ) + k X r =1 E g ( v r ) = m . Identifying ∞ in the base sphere of each Σ r with w ∗ r ∈ P , which has been already attached to 0 ∈ B ∗ ,we obtain a J -holomorphic map u ∞ : Σ ∞ −→ X with the desired properties; see Figure 12. Proof of Theorem 1.2 . Fix a Riemannian metric g Σ on Σ. For z ∈ Σ and δ ∈ Σ, let B δ ( z ) ⊂ Σdenote the ball of radius δ around z . 46 ∞ ∞ w ∗ w ∗ B Σ Σ P Xu ∞ Figure 12: Gromov’s limit of a sequence of J -holomorphic maps u i : B −→ X By Lemma 5.4, there exist a finite collection z ∗ , . . . , z ∗ ℓ ∈ Σ of distinct points and a subsequenceof { u i } , still denoted by { u i } , such that u i converges uniformly in the C ∞ -topology on compactsubsets of Σ −{ z ∗ , . . . , z ∗ ℓ } to a J -holomorphic map u : Σ −→ X . In particular, | d u i | g is uniformlybounded on compact subsets of Σ −{ z ∗ , . . . , z ∗ ℓ } . We can also assume that the limit m j ≡ lim δ −→ lim i −→∞ E g (cid:0) u i ; B δ ( z ∗ j ) (cid:1) exists for every j = 1 , . . . , ℓ . We note that E g ( u ) + ℓ X j =1 m j = lim δ −→ lim i −→∞ E g (cid:0) u ; Σ − ℓ [ j =1 B δ ( z ∗ j ) (cid:1) + ℓ X j =1 lim δ −→ lim i −→∞ E g (cid:0) u i ; B δ ( z ∗ j ) (cid:1) = lim δ −→ lim i −→∞ E g ( u i ) = lim i −→∞ E g ( u i ) . (5.22)Let δ ∈ R + be such that the balls B δ ( z ∗ i ) are pairwise disjoint. Iflim sup i −→∞ max B δ ( z ∗ j ) (cid:12)(cid:12) d u i (cid:12)(cid:12) < ∞ for some j , then { u i } converges uniformly in the C ∞ -topology on B δ ( z ∗ j ) to u by the ellipticity ofthe ¯ ∂ -operator. Thus, we can assume thatlim i −→∞ sup B δ ( z ∗ i ) (cid:12)(cid:12) d u i (cid:12)(cid:12) = ∞ for every j = 1 , . . . , ℓ .For each j = 1 , . . . , ℓ , Proposition 5.5 provides a tree Σ j of Riemann spheres P with a distinguishedsmooth point ∞ and a J -holomorphic map v j : Σ j −→ X such v j ( ∞ ) = v ( w ∗ r ) and E g ( v j ) = m j .Combining the last equality with (5.22), we obtain E g ( v ) + ℓ X j =1 E g ( v j ) = lim i −→∞ E g ( u i ) . Identifying the distinguished point ∞ of each Σ j with z ∗ j ∈ Σ, we obtain a Riemann surface (Σ ∞ , j ∞ )and a J -holomorphic map u ∞ : Σ ∞ −→ X with the desired properties.47f Σ = P and the limit map u above is constant, then ℓ ≥ ℓ ∈ { , } . Let M i = sup B δ ( z ∗ ) (cid:12)(cid:12) d u i (cid:12)(cid:12) and parametrize P so that z ∗ = 0. Define h i : P −→ P , h i ( z ) = z i + z/M i , and apply the preceding argument with u i replaced by u i ◦ h i . By the proof of Corollary 5.3(1), thelimiting map u | Σ is then non-constant and (Σ ∞ , j ∞ , u ) is a stable J -holomorphic map. We now give an example illustrating Gromov’s convergence in a classical setting.Let n ∈ Z + , with n ≥
2, and P n − = CP n − . Denote by ℓ the positive generator of H ( P n − ; Z ) ≈ Z ,i.e. the homology class represented by the standard P ⊂ P n − . A degree d map f : P −→ P n − is acontinuous map such that f ∗ [ P ] = dℓ . A holomorphic degree d map f : P −→ P n − is given by[ u, v ] −→ (cid:2) R ( u, v ) , . . . , R n ( u, v ) (cid:3) for some degree d homogeneous polynomials R , . . . , R d on C without a common linear factor.Since the tuple ( λR , . . . , λR n ) determines the same map as ( R , . . . , R n ) for any λ ∈ C ∗ , the spaceof degree d holomorphic maps f : P −→ P n − is a dense open subset of X n,d ≡ (cid:0) (Sym d C ) n − { } (cid:1)(cid:14) C ∗ ≈ P ( d +1) n − . Suppose f k : P −→ P n − is a sequence of holomorphic degree d ≥ R k = (cid:2) R k ;1 , . . . , R k ; n (cid:3) ∈ X n,d are the associated equivalence classes of n -tuples of homogeneous polynomials without a commonlinear factor. Passing to a subsequence, we can assume that [ R k ] converges to some R ≡ (cid:2) ( v u − u v ) d . . . ( v m u − u m v ) d m S , . . . , ( v u − u v ) d . . . ( v m u − u m v ) d m S n (cid:3) ∈ X n,d , (5.23)with d , . . . , d m ∈ Z + and homogeneous polynomials S ≡ [ S , . . . , S n ] ∈ X n,d without a common linear factor and with d ∈ Z ≥ . By (5.23), d + d + . . . + d m = d. Rescaling ( R k ;1 , . . . , R k ; n ), we can assume thatlim k −→∞ R k ; i = ( v u − u v ) d . . . ( v m u − u m v ) d m S i ∀ i = 1 , . . . , n. (5.24)48uppose z ∈ C − { u /v , . . . , u m /v m } . Since the polynomials S , . . . , S n do not have a commonlinear factor, S i ( z , = 0 for some i = 1 , . . . , n . This implies that R k ; i ( z , = 0 for all k largeenough and solim k −→∞ R k ; i ( z, R k ; i ( z,
1) = lim k −→∞ R k ; i ( z, k −→∞ R k ; i ( z,
1) = ( v z − u ) d . . . ( v m z − u m ) d m S i ( z, v z − u ) d . . . ( v m z − u m ) d m S i ( z,
1) = S i ( z, S i ( z, i = 1 , . . . , n and z close to z . Furthermore, the convergence is uniform on a neighborhoodof z . Thus, the sequence f k C ∞ -converges on compact subsets of P − { [ u , v ] , . . . , [ u m , v m ] } tothe holomorphic degree d map g : P −→ P n − determined by S .Let ω be the Fubini-Study symplectic form on P n − normalized so that h ω, ℓ i = 1 and E ( · ) be theenergy of maps into P n − with respect to the associated Riemannian metric. For each δ > j = 1 , . . . , m , denote by B δ ([ u j , v j ]) the ball of radius δ around [ u j , v j ] in P and let P δ = P − m [ j =1 B δ ([ u j , v j ]) . For each j = 1 , . . . , m , let m [ u j ,v j ] (cid:0) { f k } (cid:1) = lim δ −→ lim k −→∞ E (cid:0) f k ; B δ ([ u j , v j ]) (cid:1) ∈ R ≥ be the energy sinking into the bubble point [ u j , v j ]. By Theorem 1.2, the number m [ u j ,v j ] ( { f k } ) is thevalue of ω on some element of H ( P n − ; Z ), i.e. an integer. Below we show that m [ u j ,v j ] ( { f k } ) = d j .Since the sequence f k C ∞ -converges to the degree d map g : P −→ P n − on compact subsets of P −{ [ u , v ] , . . . , [ u m , v m ] } , d = h ω, d ℓ i = E ( g ) = lim δ −→ E g (cid:0) g ; P δ (cid:1) = lim δ −→ lim k −→∞ E (cid:0) f k ; P δ (cid:1) . Thus, m X j =1 m [ u j ,v j ] (cid:0) { f k } (cid:1) = m X j =1 lim δ −→ lim k −→∞ E (cid:0) f k ; B δ ([ u j , v j ]) (cid:1) = lim δ −→ lim k −→∞ E (cid:0) f k ; m [ j =1 B δ ([ u j , v j ]) (cid:1) = lim δ −→ lim k −→∞ (cid:0) E g ( f k ) − E g (cid:0) f k ; P δ (cid:1)(cid:1) = d − d = d + . . . + d m . In particular, m [ u j ,v j ] ( { f k } ) = d j if m = 1, no matter what the “residual” tuple of polynomials S is.We use this below to establish this energy identity for m > k ∈ Z + sufficiently large there exist λ k ; i ; j ; p ∈ C with i = 1 , . . . , n , j = 1 , . . . , m , and p = 1 , . . . , d j and tuples S k ≡ (cid:2) S k ;1 , . . . , S k ; n (cid:3) ∈ X n ; d of polynomials without a common linear factor such thatlim k −→∞ S k = S , lim k −→∞ λ k ; i ; j ; p = 1 ∀ i, j, p,R k ; i ( u, v ) = m Y j =1 d j Y p =1 ( v j u − λ k ; i ; j ; p u j v ) · S k ; i ( u, v ) ∀ k, i . j = 1 , . . . , m , let T j ≡ (cid:2) T j ;1 , . . . , T j ; n (cid:3) ∈ X n ; d − d j be a tuple of polynomials without a common linear factor. If in addition, i = 1 , . . . , n , ǫ ∈ R , and k ∈ Z + , let S i ; j ; ǫ ( u, v ) ≡ m Y j = j ( v j u − u j v ) d j · S i ( u, v ) + ǫT j ; i ( u, v ) , i = 1 , . . . , n,R k ; i ; j ; ǫ ( u, v ) ≡ R k ; i ( u, v ) + ǫ d j Y p =1 ( v j u − λ k ; i ; j ; p u j v ) · T j ; i ( u, v ) , i = 1 , . . . , n. The polynomials within each tuple ( S i ; j ; ǫ ) i =1 ,...,n and ( R k ; i ; j ; ǫ ) i =1 ,...,n have no common linear factorfor all ǫ ∈ R + sufficiently small and k sufficiently large (with the conditions on ǫ and k mutuallyindependent). We denote by f k ; j ; ǫ : P −→ P n − the holomorphic degree d map determined by the tuple R k ; j ; ǫ ≡ (cid:2) R k ;1; j ; ǫ , . . . , R k ; n ; j ; ǫ (cid:3) . Since lim k −→∞ R k ; j ; ǫ = (cid:2) ( v u − u v ) d j S j ; ǫ , . . . , ( v u − u v ) d j S n ; j ; ǫ (cid:3) ∈ X n ; d and the polynomials S j ; ǫ , . . . , S n ; j ; ǫ have no linear factor in common,lim δ −→ lim k −→∞ E (cid:0) f k ; j ; ǫ ; B δ ([ u j , v j ]) (cid:1) ≡ m [ u j ,v j ] (cid:0) { f k ; j ; ǫ } (cid:1) = d j (5.25)by the m = 1 case established above.For δ ∈ R + sufficiently small, ǫ ∈ R + sufficiently small, and k sufficiently large, m Y j = j d j Y p =1 ( v j u − λ k ; i ; j ; p u j v ) · S k ; i ( u, v ) = 0 ∀ [ u, v ] ∈ B δ (cid:0) [ u j , v j ] (cid:1) . Thus, the ratios R k ; i ; j ; ǫ ( u, v ) R k ; i ( u, v ) = 1 + ǫ T j ; i ( u, v ) m Q j = j d j Q p =1 ( v j u − λ k ; i ; j ; p u j v ) · S k ; i ( u, v )converge uniformly to 1 on B δ ([ u j , v j ]) as ǫ −→
0. Thus, there exists k ∗ ∈ Z + such thatlim ǫ −→ sup k ≥ k ∗ sup z ∈ B δ ([ u j ,v j ]) (cid:12)(cid:12)(cid:12)(cid:12) | d z f k ; j ; ǫ || d z f k | − (cid:12)(cid:12)(cid:12)(cid:12) = 0 . It follows that m [ u j ,v j ] (cid:0) { f k } (cid:1) ≡ lim δ −→ lim k −→∞ E (cid:0) f k ; B δ ([ u j , v j ]) (cid:1) = lim δ −→ lim k −→∞ lim ǫ −→ E (cid:0) f k ; j ; ǫ ; B δ ([ u j , v j ]) (cid:1) = lim ǫ −→ lim δ −→ lim k −→∞ E (cid:0) f k ; j ; ǫ ; B δ ([ u j , v j ]) (cid:1) = lim ǫ −→ d j = d j ;50he second-to-last equality above holds by (5.25).Suppose that either d ≥ m ≥
3. Otherwise, the maps f k can be reparametrized so that d = 0;see the last paragraph of the proof of Theorem 1.2 at the end of Section 5.3. By Theorem 1.2 andthe above, a subsequence of { f k } converges to the equivalence class of a holomorphic degree d map f : Σ −→ P n − , where Σ is a nodal Riemann surface consisting of the component Σ = P corresponding to the original P and finitely many trees of P ’s coming off from Σ . The maps onthe components in the trees are defined only up reparametrization of the domain. By the above, f | Σ is the map g determined by the “relatively prime part” S of the limit R of the tuples of poly-nomials. The trees are attached at the roots [ u j , v j ] of the common linear factors v j u − u j v of thepolynomials in R ; the degree of the restriction of f to each tree is the power of the multiplicity d j of the corresponding common linear factor.The same reasoning as above applies to the sequence of maps (cid:0) id P , f k (cid:1) : P −→ P × P n − , but the condition that either d ≥ m ≥ M , (cid:0) P × P n − , (1 , d ) (cid:1) −→ X n,d , [ f, g ] −→ (cid:2) g ◦ f − (cid:3) , from the subspace of M , ( P × P n − , (1 , d )) corresponding to maps from P extends to a continuoussurjective map M , (cid:0) P × P n − , (1 , d ) (cid:1) −→ X n,d . (5.26)In particular, Gromov’s moduli spaces refine classical compactifications of spaces of holomorphicmaps P −→ P n − . On the other hand, the former are defined for arbitrary almost Kahler manifolds,which makes them naturally suited for applying topological methods. The right-hand side of (5.26)is known as the linear sigma model in the Mirror Symmetry literature. The morphism (5.26) playsa prominent role in the proof of mirror symmetry for the genus 0 Gromov-Witten invariants in [5]and [8]; see [7, Section 30.2]. References [1] K. Behrend and B. Fantechi,
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