A Compactness Theorem for SO(3) Anti-Self-Dual Equation with Translation Symmetry
AA COMPACTNESS THEOREM FOR SO p q ANTI-SELF-DUALEQUATION WITH TRANSLATION SYMMETRY
GUANGBO XU
Abstract
Motivated by the Atiyah–Floer conjecture, we consider SO p q anti-self-dual instantons on theproduct of the real line and a three-manifold with cylindrical end. We prove a Gromov–Uhlenbecktype compactness theorem, namely, any sequence of such instantons with uniform energy boundhas a subsequence converging to a type of singular objects which may have both instanton andholomorphic curve components. This result is the first step towards constructing a naturalbounding cochain proposed by Fukaya for the SO p q Atiyah–Floer conjecture.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. The rescaled equation and interior compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194. The isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235. Compactness modulo energy blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356. Stable scaled instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407. Proof of the compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.
Introduction
The Atiyah–Floer conjecture.
In 1980s Floer [Flo88a][Flo88b][Flo89] introducedseveral important invariants of different types of geometric objects. These invariants arenow generally called the Floer (co)homology. The Atiyah–Floer conjecture [Ati88] assertsthat two such invariants, the instanton Floer homology of a three-dimensional manifold andthe
Lagrangian intersection Floer homology associated to a splitting of the three-manifold,are isomorphic. This conjecture has become a central problem in the field of symplecticgeometry, gauge theory, and low-dimensional topology. The principle underlying theAtiyah–Floer conjecture has also motivated a number of important constructions inlow-dimensional topology, such as the Heegaard–Floer homology [OS04]. There are twoprincipal versions of the Atiyah–Floer conjecture, the SU p q case and the SO p q case,corresponding respectively to the two choices of the gauge group. Despite many progressesmade in recent years, the general cases of both versions are still open. On the level ofEuler characteristics, the Atiyah–Floer conjecture was proved by Taubes [Tau90].Let us briefly review Atiyah’s intuitive argument [Ati88] leading to the identification ofthe two Floer homologies. Let M be a closed oriented three-manifold. Let Σ Ă M be anembedded surface separating M into two pieces M ´ and M ` which share the commonboundary Σ (see Figure 1.1). Consider a G -bundle P Ñ M where G is either SU p q or SO p q . The moduli space of flat connections on P | Σ , denoted by R Σ , is naturally a Date : June 23, 2020. a r X i v : . [ m a t h . S G ] J un OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 2 (singular) symplectic manifold. Inside R Σ there are two Lagrangian submanifolds L M ´ and L M ` associated to the splitting, i.e., the set of gauge equivalence classes of flat connectionson Σ which can be extended to a flat connection on P | M ` resp. P | M ´ . The generators ofthe instanton Floer chain complex, which are gauge equivalence classes of flat connectionson M , correspond naturally to intersections of the two Lagrangian submanifolds. Theseintersection points are generators of the Lagrangian Floer chain complex. On the otherhand, the differential map of the instanton Floer homology I p M, P q is defined by countingsolutions to the anti-self-dual equation (the ASD equation) on the product R ˆ M of thereal line R and M . This equation depends the metric on M while the resulting homologyis independent of the metric. If one “stretches the neck,” namely one chooses a familyof metrics g T on M such that a fixed neighborhood of Σ is isometric to r´ T, T s ˆ
Σ,then one expects that solutions to the ASD equation converge as T Ñ 8 to holomorphicmaps u : r´ , s ˆ R Ñ R Σ with boundary condition u pt˘ u ˆ R q Ă L M ˘ —the countingof these holomorphic maps defines the differential map of the Lagrangian intersectionFloer chain complex with resulting homology group HF p L M ´ , L M ` q . The correspondencebetween instantons and holomorphic strips shows that the two Floer chain complexes,and hence the homology groups, should be isomorphic. M − M + Σ M − M + [ − T, T ] × Σ Figure 1.1.
Splitting and neck-stretching along an embedded surface.This paper is motivated by the SO p q case of the Atiyah–Floer conjecture. We firstremark on a crucial difference between the SO p q case and the SU p q case. When G “ SU p q , the moduli space of flat connections over a surface has singularities correspondingto reducible connections. This fact makes the Lagrangian Floer homology HF p L M ´ , L M ` q difficult to define (see [MW12] for an equivariant construction of this Floer homology).When G “ SO p q , the moduli space R Σ of flat connections on a nontrivial SO p q -bundleover a surface is smooth. Consequently the Lagrangian Floer homology can be definedin the traditional way and the corresponding Atiyah–Floer conjecture has been provedin certain cases. For example, Dostoglou–Salamon [DS94] proved the SO p q case formapping cylinders.Another motivation of this paper comes from the original neck-stretching argumentsketched above. As people become more interested in the alternative approach usingLagrangian boundary conditions for the instanton equation (see [Sal95] [Fuka, Fuk98][Weh05a, Weh05b, Weh05c] [SW08] [Fukb] [DF18]), the neck-stretching argument ismore or less abandoned. Furthermore, the neck-stretching argument can provide adirect comparison between the moduil spaces, potentially leading to Atiyah–Floer typecorrespondences for more refined invariants. We would like to see how far Atiyah’s originalidea can go beyond the situation of [DS94]. Meanwhile, the analytic problems involved inthe neck-stretching limit have their own interests and deserve to be explored. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 3
Bounding cochain on the symplectic side.
A more direct motivation of thispaper is a conjecture of Fukaya [Fuk18, Conjecture 6.7]. This conjecture is related to anadditional complexity in the study of the Atiyah–Floer conjecture, that is, the regularity ofthe Lagrangian submanifolds. From now on we assume G “ SO p q , which grants a smoothmoduli space R Σ of flat connections on the nontrivial SO p q -bundle over the surfaceΣ. If both Lagrangians L M ` and L M ´ are embedded and intersect transversely, thenone can define HF p L M ´ , L M ` q in a standard way as in [Flo88b] [Oh93a, Oh93b], as thetwo Lagrangians are both monotone. In general, however, the natural maps L M ˘ Ñ R Σ are not embeddings. After a generic perturbation, one can only achieve immersions L M ˘ í R Σ which satisfy the monotonicity condition in a weak sense. In this situation weneed to apply the more complicated construction of immersed Lagrangian Floer homologydeveloped by Akaho–Joyce [AJ10] with the appearance of bounding cochains . A generalLagrangian immersion may not have bounding cochains; even it has, the Floer homologyinvolving the immersed Lagrangian depends on the choice of a bounding cochain.The necessity of considering bounding cochains in the Atiyah–Floer conjecture canalso be seen via a closer look at the neck-stretching process. As one stretches the neck,the energy of ASD instantons can be distributed in three parts, R ˆ M ´ , R ˆ M ` , and R ˆ r´ T, T s ˆ
Σ (corresponding to the left, the right, and the middle parts of the secondpicture of Figure 1.1). The energy stored on the left and on the right can form ASDinstantons on R ˆ M ´8 and R ˆ M `8 , where M ˘8 is the completion of M ˘ by addingthe cylindrical end Σ ˆ r , `8q . On the other hand, the energy stored in the middlepart produces holomorphic strips in R Σ as predicted by the Atiyah–Floer conjecture.Hence a general limiting object could be a complicated configuration having componentscorresponding to either instantons over R ˆ M ˘8 or holomorphic strips (see Figure 1.2); weignore bubbling of instantons over R , instantons over C ˆ Σ, and holomorphic spheres, asthey happen in high codimensions. While in the case when L M ˘ are immersed, instantonsover R ˆ M ˘8 may give nontrivial contributions which happen in codimension zero. L M − L M + u A A A A A Figure 1.2.
A typical limiting configuration in the adiabatic limit: u is a holomorphic strip in R Σ with boundary in L M ´ and L M ` , A and A are ASD instantons over R ˆ M ´8 , A , A , A are ASD instantonsover R ˆ M `8 . The limits of the instantons A i are double points of theLagrangian immersions L M ˘ .The geometric picture discussed above suggests that one has to modify the LagrangianFloer chain complex to match with the instanton Floer chain complex. The differentialmap of the modified chain complex counts not only usual holomorphic strips, but also OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 4 strips with additional boundary constraints associated with the instantons over R ˆ M ˘8 .This kind of boundary constraints can be regarded as bounding cochains. As introduced in[FOOO09], for any cochain model chosen for the Lagrangian Floer theory (such as Morsecochains), a bounding cochain is a cochain of odd degree which can cancel all contributionsof disk bubbling; these disk bubbles may obstructed d “ L ´ , L ` are Lagrangian submanifolds, then a choiceof a pair of bounding cochains b ´ , b ` leads to a deformed complex CF pp L ´ , b ´ q , p L ` , b ` qq .The differential map of the deformed complex counts holomorphic strips with boundary“insertions,” i.e., points on the two boundary components satisfying geometric constraintsprescribed by the cochains b ´ and b ` (see Figure 1.3). p qb − b − · · · b − b − · · · b + b + b + Figure 1.3.
A holomorphic strip with boundary insertions lying in con-straints given by the bounding cochains.The following conjecture of Fukaya summarizes the above discussion and directlymotivates the work of the current paper.
Conjecture 1.1. [Fuk18, Conjecture 6.7]
Let M be a three-manifold with a cylindricalend isometric to Σ ˆ r , `8q and P Ñ M be an SO p q -bundle whose restriction to everyconnected component of Σ is nontrivial. Let L M be the moduli space of flat connectionsover M . Suppose the natural map L M Ñ R Σ is an immersion with transverse doublepoints. Then “counting” instantons on R ˆ M defines a bounding cochain b M P CF p L M q . of the A -algebra associated to the immersed Lagrangian L M í R Σ . Indeed, for the situation of immersed Lagrangians considered in [AJ10], the cochainmodel has summands corresponding to the double points of the immersion. Hence apriori a bounding cochain is in general a linear combination of ordinary cochains on L M and double points of the immersion. As observed by Fukaya [Fuk18], due a weak versionof monotonicity, the bounding cochain b M in the above conjecture is expected to be alinear combination of only double points.A refined version of the SO p q Atiyah–Floer conjecture can be stated as follows.
Conjecture 1.2 (The SO p q Atiyah–Floer conjecture) . For a closed three-manifold M with a suitable SO p q -bundle P Ñ M and a suitable splitting M “ M ´ Y M ` , there is anatural isomorphism of abelian groups I p M, P q – HF pp L M ´ , b M ´ q , p L M ` , b M ` qq . (1.1) OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 5
Remark . In [Fukb, Fuk18] a different strategy of proving the existence of a boundingcochain was sketched. Instead of considering instantons over R ˆ M where M has acylindrical end, consider the ASD equation over R ˆ M where M is the correspondingmanifold with boundary, imposing the boundary condition given by a testing Lagrangian L Ă R Σ . The advantage of this approach is that it can avoid certain difficult analysisassociated to the ASD equation on R ˆ M , while a simpler moduli space is enough toproduce a chain map. However, this approach lacks a direct comparison between themoduli space of instantons and moduli space of holomorphic strips as indicated by thestraightforward neck-stretching argument. Such a comparison between moduli spacescan be useful in establishing relations between more refined invariants. The approach ofusing Lagrangian boundary conditions have also been adopted to solve the Atiyah–Floerconjecture, see [Sal95] [Fuka, Fuk98, Fukb] [Weh05c, Weh05a, Weh05b] [SW08] [DF18].1.3. Main results of this paper.
The purpose of this paper is to take the first steptowards the resolution of Conjecture 1.1, namely, to compactify the moduli space of ASDinstantons over R ˆ M where M is a three-manifold with cylindrical end. More precisely,given a sequence of ASD instantons A i over R ˆ M with uniformly bounded energy,we study the possible limiting configurations as i Ñ 8 . There are several phenomenapreventing A i from converging to an ASD instanton. (i) As in the usual situation of theASD equation, energy may concentrate in small scales and bubble off instantons on R .(ii) Since R ˆ M is noncompact, the energy may concentrate at different regions of thesame scale which move apart from each other. This is similar to the situation in Morsetheory, where a sequence of gradient lines can converge to a broken gradient line. Therecan also be instantons over C ˆ Σ appearing as energy may escape in the noncompactdirection of M . (iii) A nontrivial amount of energy may escape from any finite regionof R ˆ M and spread over larger and larger domains; after rescaling such energy formeither holomorphic spheres or holomorphic disks in R Σ . In general a combination ofthese phenomena can happen in the limit. The hierarchy of different speeds of energyconcentration or spreading is captured by the combinatorial type of the limiting objectdescribed by a tree. See Figure 1.4 for a typical configuration of the limiting object, whichwe will call stable scaled instantons.One can see that the limiting configurations are very similar to objects appearing inthe adiabatic limit of the symplectic vortex equation (see [GS05] and [Zil05, Zil14] forthe closed case and [WX17] [WX] for the case with Lagrangian boundary condition).Combinatorially these objects are also similar to certain objects appearing in the compact-ification of pseudoholomorphic quilts studied by [WW15] and [BW18]. To describe suchlimiting objects, we need to define certain singular configurations which have componentscorresponding to energy concentrations in different scales (see [WX17, Section 4] andSection 6 of the current paper). Having this picture in mind, in this paper we define thenotion of stable scaled instantons (see Definition 6.2) as the expected limiting objects,and a Gromov–Uhlenbeck type convergence (see Definition 6.3). Then we can state ourmain theorem as follows. Theorem 1.4.
Let M be a three-manifold with cylindrical end and P Ñ M be an SO p q -bundle. Suppose p M, P q satisfies assumptions of Conjecture 1.1. Then given a sequenceof anti-self-dual instantons on R ˆ P Ñ R ˆ M with uniformly bounded energy, there isa subsequence which converges modulo gauge transformation and translation to a stablescaled instanton (in the sense of Definition 6.3). Besides proving the above compactness theorem, using the same method and moresimplified argument, we can also prove a counterpart for instantons over C ˆ Σ. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 6
Figure 1.4.
A typical stable scaled instanton with eight components.The dark regions are holomorphic curve components while the corners arewhere the curves meet the double points of immersed Lagrangian. Thewhite regions are instanton components: the two on the left are instantonsover R ˆ M and the one on the right (the teardrop) is an instanton over C ˆ Σ. Theorem 1.5.
Let Σ be a compact Riemann surface (not necessarily connected) and Q Ñ Σ be an SO p q -bundle which is nontrivial over every connected component of Σ .Then given a sequence of anti-self-dual instantons on C ˆ Q Ñ C ˆ Σ with uniformlybounded energy, there is a subsequence which converges modulo gauge transformation andtranslation to a stable scaled instanton. We remark that certain compactness problems in gauge theory with respect to adiabaticlimit or neck-stretching which are of similar nature have been considered by other people,for example Chen [Che98], Nishinou [Nis10], and Duncan [Dun13, Dun]. Comparingto these previous works, the main contribution of this paper is the treatment of thecompactness problem near the “boundary” (the compact part of the three-manifoldwith cylindrical ends). The argument is based on the isoperimetric inequality (Theorem4.4), the annulus lemma (Proposition 4.5 and extensions), and the boundary diameterestimate (Lemma 4.10). The method of using boundary diameter estimate to establishboundary compactness would also be useful in other situations. For example, for thecompactness problem about the strip-shrinking limit of pseudoholomorphic quilts, thismethod potentially leads to a simplified argument as opposed to the method of [BW18]which appeals to hard elliptic estimates over varying domains. Another contribution ofthe current paper is to define the correct notion of singular configurations (stable scaledinstantons) that may appear in the limit and provide detailed argument of constructingthe limiting bubble tree. Last but not the least, a modification of our construction willlead to a proof of a compactness theorem about the neck-stretching limit for the SO p q instanton equation. The details will be completed in future works.This paper is organized as follows. In Section 2 we recall basic notions and factsabout the anti-self-dual equation, holomorphic curves, state the main assumption ofthe three-manifold, and recall a few technical results. In Section 3 we recall a basiccompactness theorem for the rescaled ASD equation over the product of two surfaces inthe adiabatic limit and prove a refinement of an interior estimate of Dostoglou–Salamon. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 7
In Section 4 we (re)prove an isoperimetric inequality for a closed three-manifold and theannulus lemma, and establish a boundary diameter estimate. In Section 5 we prove thecompactness modulo energy blowup theorem for the ASD equation over the noncompactfour-manifold R ˆ M . In Section 6 we state the main theorem in technical terms. InSection 7 we finish the proof of the main theorem (Theorem 1.4 and Theorem 1.5). Acknowledgments.
The author would like to thank Kenji Fukaya for suggesting thisproblem, for many stimulating discussions, and for his warm encouragement and generoussupport. The author would like to thank Simons Center for Geometry and Physics forcreating a wonderful environment for mathematical research. The author would like tothank Donghao Wang and David Duncan for helpful discussions.This work is partially supported by the Simons Collaboration Grant on HomologicalMirror Symmetry. 2.
Preliminaries
In this paper we study gauge theory for a smooth SO p q -bundles P Ñ U where U is some smooth manifold of dimension at most four. As in the usual treatmentof SO p q -gauge theory, we modify the definition of gauge transformations as follows.The conjugation of SO p q can be extended to an SO p q -action on SU p q . A gaugetransformation on P is regarded as SU p q -valued, i.e., a map g : P Ñ SU p q satisfying g p ph q “ h ´ g p p q h, @ h P SO p q , p P P. Since so p q – su p q , such SU p q -valued gauge transformations acts on SO p q -connectionsin the usual way. Let A p P q be the space of smooth connections on P and G p P q thespace of SU p q -valued smooth gauge transformations. The gauge equivalence class of aconnection A P A p P q is usually denoted by r A s .In this paper, when there is no extra explanation, the sequential convergence of smoothobjects are always regarded as convergence in the C loc -topology.2.1. Chern–Simons functional and the anti-self-dual equation.
The Chern–Simons functional.
The instanton Floer cochain complex can be formallyviewed as the Morse cochain complex for the Chern–Simons functional. Let M be asmooth oriented four-manifold. Let P Ñ M be a smooth SO p q -bundle. By Chern–Weiltheory, the Pontryagin class can be represented by the differential form p p A q “ ´ π tr p F A ^ F A q P Ω p M q for any smooth connection A on P . When M is closed, the integral of this differentialform is an integer and is a topological invariant. On the other hand, suppose M has anonempty boundary B M – M where M inherits a natural orientation. Denote P “ P | M .Then for any smooth connection A P A p P q , for any smooth extension A of A to theinterior, the Chern–Weil integral12 ż M tr p F A ^ F A q P R only depends on the gauge equivalence class of A . This is called the Chern–Simonsfunctional , denoted by CS P : A p P q{ G p P q Ñ R . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 8
Often we will omit the dependence on the bundle P from the notation. When M has several connected components M , . . . , M k and r A s , . . . , r A k s are the restrictions of r A s P A p P q{ G p P q to these components, we denote this action by CS pr A s , . . . , r A k sq . In particular, for a three-manifold N with an SO p q -bundle P Ñ N , for a referenceconnection A and any A P A p P q , denote the relative Chern–Simons action of A to be CS rel r A s pr A sq “ CS P pr A s , r A sq where P “ r , s ˆ P is the product bundle over r , s ˆ N . When A is flat, the action isequal to ż M tr ” d A p A ´ A q ^ p A ´ A q ` p A ´ A q ^ p A ´ A q ^ p A ´ A q ı . (2.1)This coincides with the classical expression of the Chern–Simons functional.The Chern–Simons functional can be viewed as a multivalued function on the space ofconnections on the three-manifold. For each A P A p P q and a deformation α P Ω p ad P q ,the directional derivative of the Chern–Simons functional in the direction of α is D A CS p α q “ ż M tr p F A ^ α q . Hence critical points of the Chern–Simons functional are flat connections.2.1.2.
The anti-self-dual equation.
Suppose M is equipped with a Riemannian metric.A connection A P A p P q is called an anti-self-dual connection (ASD connection forshort ) if F A ` ˇ F A “ P Ω p M , ad P q . Here F A is the curvature of A and ˇ is the Hodge start operator on differential forms on M . When M is noncompact, we also impose the finite energy condition. Then definethe Yang–Mills functional , also called the energy , of a connection A by E p A q : “ ż M | F A | . Here the norm of the curvature is induced from the metric on M and the Killing metricon the Lie algebra. We always assume that ASD connections have finite energy.A particular case is when M “ R which is equipped with the standard Euclideanmetric. We call an ASD connection over R an R -instanton .The ASD equation can be viewed as the gradient flow equation of the Chern–Simonsfunctional. A direct consequence of this perspective is the following energy identity. Lemma 2.1.
Let M be a compact oriented four-manifold with boundary and P Ñ M bean SO p q -bundle. Then for any Riemannian metric on M and any ASD connection on P with respect to this metric, one has E p A q “ CS pr A | B M sq . When the domain is R , or the product of the complex plane and a closed surface, or the product ofthe real line and a three-manifold with cylindrical end, we call an ASD connection an instanton. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 9
Convergence and compactness.
We discuss the topology of the space of connectionsand recall the celebrated Uhlenbeck compactness theorem. Let M be a manifold and P Ñ M be a principal bundle. The convergence of smooth SO p q -connections A i P A p P q towards a limit A P A p P q (in the C loc topology) means for any precompact open subset K Ă M , A i | K converges uniformly with all derivatives to A | K . We define the moregeneral notion of convergence in the Uhlenbeck sense. Let M be a Riemannian four-manifold. Let M i Ă M be an exhausting sequence of open subsets, meaning that everycompact subset of M is contained in M i for sufficiently large i . Let P i Ñ M i be SO p q -bundles and A i P A p P i q be a sequence of ASD connections on P i . Let P Ñ M be an SO p q -bundle and A P A p P q be an ASD connection on P . Let m be a positivemeasure on M supported at finitely many points. Definition 2.2.
We say that A i converges to p A , m q in the Uhlenbeck sense if(a) the sequence of functions | F A i | converge as measures to | F A | ` m , and(b) there are bundle isomorphisms ρ i : P Ñ P i over M i (cid:114) Supp m such that ρ ˚ i A i converges to A .This notion of convergence is independent of the choices of representatives in theirgauge equivalence classes. Therefore if A i and p A , m q satisfy conditions of Definition2.2, we will say that r A i s converges to pr A s , m q in the Uhlenbeck sense. Further, themeasure m in the limit, called the bubbling measure , is nonzero at x P M if andonly if a nontrivial R -instanton bubbles off in the limit. We know that the masses of m are in 4 π Z ` . The convergence implies the following energy identity: for any compactsubset K Ă M containing the support of m , there holdslim i Ñ8 E p A i ; K q “ E p A ; K q ` ż K m . We summarize the celebrated Uhlenbeck compactness theorem as follows.
Theorem 2.3. ( cf. [DK90, Section 4.4]) Let M , M i , P i , A i be as above. Suppose theenergy of A i is uniformly bounded from above, i.e, lim sup i Ñ8 E p A i q ă `8 . Then there exist a subsequence (still indexed by i ), a positive measure m on M withfinite support, an SO p q -bundle P Ñ M , and an ASD connection A P A p P q , suchthat A i converges to p A , m q in the Uhlenbeck sense. In particular, there is a positiveconstant (cid:126) ą with the following property: if E p A i q ă (cid:126) , then a subsequence of A i converges to a limiting ASD connection A without bubbling. Product of two surfaces.
The Atiyah–Floer conjecture can be regarded as aneffect of the reduction from 4D gauge theory to 2D sigma model observed by physicists[BJSV95]. Consider the special case that M “ S ˆ Σ where S and Σ are oriented surfaces,equipped with a product metric. We assume for simplicity that S is an open subset ofeither the complex plane C or the upper half plane H , in which cases S is equippedwith the standard holomorphic coordinate z “ s ` i t and the standard flat metric. Let Q Ñ Σ be an SO p q -bundle and P “ S ˆ Q Ñ M be the pullback of Q via the projection M Ñ Σ. Then we can write a connection A P A p P q as A “ d S ` φds ` ψdt ` B, where d S is the exterior differential in S , B : S Ñ A p Q q is a smooth map, and φ, ψ P Ω p S ˆ Σ , ad Q q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 10
Now we look at the ASD equation with respect to the product metric. Introduce A s “ B s B ´ d B φ, A t “ B t B ´ d B ψ. (2.2)and κ A “ B s ψ ´ B t φ ` r φ, ψ s , µ A “ F B . (2.3)Then with respect to the product metric, the ASD equation can be written in the localform A s ` ˚ A t “ , κ A ` ˚ µ A “ . (2.4)Here ˚ is the Hodge star on Σ. Remark . The equation (2.4) can be viewed as an infinite dimensional version of thesymplectic vortex equation introduced by Cieliebak–Gaio–Salamon [CGS00] and Mundet[Mun99, Mun03]. Indeed this perspective is one of the motivation of [CGS00] to proposethe symplectic vortex equation.2.3.
Holomorphic curves.
We recall a few basic facts about pseudoholomorphic mapsfrom Riemann surfaces to almost complex manifolds. In this subsection, p X, J q alwaysdenotes a compact almost complex manifold. We fix a Riemannian metric h on X . Let S be a smooth Riemann surface with possibly nonempty boundary. A J -holomorphicmap from S to X is a continuous map u : S Ñ X which is smooth in the interior andsatisfies the Cauchy–Riemann equation B J u : “ ˆ B u B s ` J B u B t ˙ d ¯ z “ . Here z “ s ` i t is a local holomorphic coordinate on S . In this paper S is always an opensubset of either the complex plane C or the upper half plane H . The energy of u is E p u ; S q : “ } du } L p S q “ ż S | du | h dsdt. When S is understood from the context, we abbreviate E p u ; S q by E p u q .2.3.1. Compactness.
We would like to define the notion of convergence of J -holomorphicmaps over a bordered surface without any appropriate boundary condition. Let S Ă H be an open subset and S i Ă S be an exhausting sequence of open subsets. Definition 2.5.
Let u i : S i Ñ X be a sequence of J -holomorphic maps. Let u : S Ñ X be another J -holomorphic map. Then we say that u i converges to u if u i converges to u in C p S q X C loc p S X Int H q .Now we recall a compactness result about holomorphic maps on bordered surfaceswithout imposing a boundary condition and give a proof. Proposition 2.6.
Let u i : S i Ñ X be a sequence of J -holomorphic maps with lim sup i Ñ8 E p u i ; S i q ă `8 . (a) Assume there is no energy concentration in the interior, namely, for all z P S X Int H , one has lim r Ñ lim sup i Ñ8 E p u i ; B r p z q X S i q “ . (2.5) Then there exist a subsequence (still indexed by i ) and a holomorphic map u : S X Int H Ñ X such that u i converges to u in C loc p S X Int H q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 11 (b) In addition, suppose for each z P B S there holds lim r Ñ lim sup i Ñ8 diam p u i p B ` r p z q X S i qq “ (here B ` r p z q is the half disk B r p z q X H ). Then the limit u extends continuouslyto S X B H and u i converges to u in the sense of Definition 2.5.Proof. Part (a) is the classical Gromov compactness result (see for example [MS04] [IS00]).For part (b), we first show that u has limits at all boundary points. Choose z P B S . By(2.6), for any (cid:15) ą
0, there exists an r ą i Ñ8 diam p u i p B ` r p z qqq ă (cid:15). Then for sufficiently large i , for all z , z P B ` r p z q X S X Int H , there holds d p u i p z q , u i p z qq ă (cid:15). Since u i p z q Ñ u p z q and u i p z q Ñ u p z q as i Ñ 8 . It implies that z , z P B ` r p z q ùñ d p u p z q , u p z qq ă (cid:15). Hence u has limits at all boundary points. It is a similar argument to show that theboundary limits define a continuous extension of u and u i converges to u in C p S q .We leave the details to the reader. (cid:3) Immersed Lagrangian boundary condition.
We recall basic notions of pseudoholo-morphic curves with an immersed Lagrangian boundary condition. We fix a compactsymplectic manifold p X, ω q . A Lagrangian immersion is a smooth immersion ι : L í X such that ι ˚ ω “ X “ L . We assume L is compact. We assumethat L only has transverse double points . Namely, the map p ι, ι q : L ˆ L Ñ X ˆ X is transverse to the diagonal ∆ X Ă X ˆ X away from the diagonal ∆ L Ă L ˆ L . Thecompactness of L implies that there are finitely many double points of ι p L q , each ofwhich has exactly two preimages p p, q q , p q, p q P L ˆ L (cid:114) ∆ L . Elements of the set R L : “ tp p, q q P L ˆ L | ι p p q “ ι p q q , p ‰ q u . are called ordered double points . The map p p, q q ÞÑ p q, p q which preserves the set∆ L Y R L is called the transpose .Now we define the notion of holomorphic curves with boundary lying in the immersedLagrangian. One can see that this notion coincides with that in [AJ10] after ordering theset of marked points W . Fix an ω -tamed almost complex structure J on p X, ω q , namely ξ ÞÑ ω p ξ, J ξ q ą , @ ξ P T X, ξ ‰ . The symplectic form ω and the almost complex structure J determines a metric g p ξ, ξ q : “ ` ω p ξ, J ξ q ` ω p ξ , J ξ q ˘ . We also assume that L is totally real with respect to J , namely, for every x P L , T ι p x q X “ dι p T x L q ‘ J dι p T x L q . Definition 2.7. (cf. [AJ10, Definition 4.2]) Let S be a Riemann surface with possiblynonempty boundary. A marked J -holomorphic map from S to X with boundary in ι p L q is a quadruple u “ p u, W, γ q OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 12 where u : S Ñ X is a J -holomorphic map with u pB S q Ă ι p L q , W Ă S is a finite subset,and γ : B S (cid:114) W Ñ L is a smooth map, satisfying the following boundary condition : u | B S (cid:114) W “ ι ˝ γ. Given such a marked J -holomorphic map, for each w P W X B S , there is a localholomorphic coordinate chart U w – φ w p U w q Ă H . The boundary condition implies thatthe limit ev w p u q : “ p lim s Ñ ´ γ p φ ´ w p s qq , lim s Ñ ` γ p φ ´ w p s qqq P ∆ L Y R L Ă L ˆ L exists. We call this limit the evaluation of u at w . If ev w p u q P R L , we call w a switchingpoint of u . On the other hand, the evaluation of u at an interior marking w P W X Int S is the value ev w p u q “ u p w q P X .We also include a nontrivial Dirac measure in the datum of a holomorphic curve.If m : W Ñ R ` is a function, regarded as a positive measure on S whose support iscontained in W , then we call the tuple˜ u “ p u, W, γ, m q a marked holomorphic curve with mass . When B S “ H , we simplify the notationas ˜ u “ p u, W, m q .2.4. Flat connections on three-manifolds.
Now we introduce the basic assumptionson the three-manifolds. Let M be a connected, oriented three-manifold with a nonemptyand not necessarily connected boundary B M – Σ. Let M be the completion, i.e., M : “ M Y pr , `8q ˆ Σ q , (2.7)where the two parts are glued along the common boundary. Then we always identify M with a closed subset of M . Let P Ñ M be an SO p q -bundle. Let P Ñ M be therestriction of P to M Ă M and Q Ñ Σbe the restriction of P to the boundary. Let L M be the moduli space of gauge equivalenceclasses of flat connections on P , i.e., L M : “ ! A P A p P q | F A “ ) { G p P q . Let R Σ be the moduli space of gauge equivalence classes of flat connections on Q , i.e., R Σ : “ ! B P A p Q q | F B “ ) { G p Q q . Both L M and R Σ have natural topology. There is a natural continuous map ι : L M Ñ R Σ induced by boundary restriction.2.4.1. Transversality assumption.
Now we consider moduli spaces of flat connections on M and Σ. We impose certain extra conditions to guarantee that these moduli spaces aresmooth. For any flat connection A P A p P q , the covariant derivative d A makes ad P aflat bundle with a twisted de Rham complex0 (cid:47) (cid:47) Ω p M , ad P q d A (cid:47) (cid:47) Ω p M , ad P q d A (cid:47) (cid:47) Ω p M , ad P q d A (cid:47) (cid:47) Ω p M , ad P q (cid:47) (cid:47) . Similarly, when B is a flat connection on Q , there is a complex0 (cid:47) (cid:47) Ω p Σ , ad Q q d B (cid:47) (cid:47) Ω p Σ , ad Q q d B (cid:47) (cid:47) Ω p Σ , ad Q q (cid:47) (cid:47) . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 13
When A | Σ “ B , one can form the relative complex withΩ k rel p ad P q “ Ω k p M , ad P q ‘ Ω k ´ p Σ , ad Q q and differential d A,B , which is defined as d A,B « αh ff “ « d A ´ r ˚ d B ff « αh ff . Here r ˚ : Ω k p M , ad P q Ñ Ω k p Σ , ad Q q is the pullback. Then there is a long exact sequence(in real coefficients) ¨ ¨ ¨ (cid:47) (cid:47) H p d A q (cid:47) (cid:47) H p d B q (cid:47) (cid:47) H p d A,B q (cid:47) (cid:47) H p d A q (cid:47) (cid:47) H p d B q (cid:47) (cid:47) ¨ ¨ ¨ We assume the following conditions throughout this paper.
Hypothesis . The three-manifold with boundary M and the SO p q -bundle P Ñ M satisfy the following conditions.(a) For any flat connection B on Q , H p d B q ‘ H p d B q vanishes. This implies that R Σ is a smooth manifold with dim R Σ “ ´ χ p Σ q . (b) For any flat connection A on P , the operator d A : Ω p M , ad P q Ñ Ω p M , ad P q is surjective. This implies that the moduli space L M of flat connections on P issmooth and dim L M “ ´ χ p Σ q . (c) For any flat connection A on P whose boundary restriction is B , the map H p d A q Ñ H p d B q is injective. This implies that the natural map L M Ñ R Σ isan immersion.(d) The immersion ι : L M í R Σ has transverse double points. Lemma 2.9.
Item (a) of Hypothesis 2.8 holds if and only if Q is nontrivial over eachconnected component of Σ . In this case Σ necessarily has an even number of connectedcomponents.Proof. If Q is trivial over some component Σ i Ă Σ, then there exist reducible flatconnections, i.e., there are flat connections B P A flat p Q q with H p d B q ‰
0. On the otherhand, one has the Poincar´e duality H p d B q – H p d B q . Hence (a) is equivalent to thenontriviality of Q over each Σ i .To show that Σ has an even number of connected components, consider the exactsequence in Z coefficients ¨ ¨ ¨ (cid:47) (cid:47) H p M q (cid:47) (cid:47) H pB M q (cid:47) (cid:47) H p M , B M q (cid:47) (cid:47) ¨ ¨ ¨ It follows that the second Stiefel–Whitney class w p Q q , which is the image of w p P q P H p M q , is sent to zero in H p M , B M q – Z . On the other hand, since Q is nontrivialover each component Σ i , w p Q q restricts to the generator of H p Σ i ; Z q , while eachgenerator is sent to the generator of H p M , B M q – Z . Hence Σ has an even number ofconnected components. (cid:3) Remark . In general Item (b), (c), and (d) of Hypothesis 2.8 do not hold. However,one can perturb the Chern–Simons functional by the so-called holonomic perturbationsupported away from the boundary, so that critical points of the Chern–Simons functional(i.e., certain perturbed flat connection) are non-degenerate in the Bott sense. Let L M still denote the moduli space of critical points of the perturbed Chern–Simons functional. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 14
Pairs of flat connections on two three-manifolds with boundary induce certain flatconnections on a closed three-manifold. Let N ´ , N ` be connected oriented three-manifoldswith boundary such that B N ´ – Σ – Bp N ` q op . Here p N ` q op is a copy of N ` with the reversed orientation. Let P N ´ Ñ N ´ , P N ` Ñ N ` ,and Q Ñ Σ be SO p q -bundles such that P N ´ | B N ´ – Q – P N ` | B N ` . Suppose p P N ´ , N ´ q and p P N ` , N ` q both satisfy Hypothesis 2.8. Then one obtainsLagrangian immersions ι ´ : L N ´ í R Σ , ι ` : L N ` í R Σ . We assume in addition that
Hypothesis . The two three-manifolds with boundary N ´ , N ` with diffeomorphicboundary and the SO p q -bundles P N ˘ Ñ N ˘ with isomorphic boundary restrictionssatisfy the following condition. ‚ The immersions ι ´ : L N ´ í R Σ and ι ` : L N ` í R Σ intersect cleanly.Define a closed three-manifold and together with an SO p q -bundle as follows. Let N be the closed three-manifold defined by N : “ N ´ Y N neck Y N ` : “ N ´ Y pr´ , s ˆ Σ q Y N ` . (2.8)Here we identify the common boundaries of the two components via B N ´ – t´ u ˆ Σ, Bp N ` q op – t u ˆ Σ. The bundles P N ˘ can be glued similarly to give an SO p q -bundle P N : “ P N | N ´ Y P N | N neck Y P N | N ` : “ P N ´ Y pr´ , s ˆ Q q Y P N ` . (2.9)Let L N be the moduli space of gauge equivalence classes of flat connections on P N . Then L N is a compact manifold (with possibly varying dimensions) with a diffeomorphism L N – p ι ´ ˆ ι ` q ´ p ∆ R Σ q Ă L N ´ ˆ L N ` . Remark . There are two special situations when Hypothesis 2.11 is satisfied. Thefirst special case is when N ´ – p N ` q op – M and P N ´ – P N ` – M . In this case N isdiffeomorphic to the doubling of M , denoted by M double and one has L N – ∆ L M Y R L M Ă L M ˆ L M . The second special case is when N ´ – p N ` q op – r , π s ˆ Σ whose boundary is two copiesof Σ and P N ´ – P N ` – r , π s ˆ Q . In this case N is diffeomorphic to S ˆ Σ and L N isdiffeomorphic to R Σ .It is convenient to allow certain piecewise smooth connections. Define A p . s . p P N q : “ tp A ´ , A , A ` q P A p P N | N ´ \ P N | N neck \ P N | N ` q | p A ´ , A ` q| B N ´ YB N ` “ A | B N neck u whose elements are called piecewise smooth connections . Define the space of piecewisesmooth gauge transformations G p . s . p P N q in a similar way. Then one has L N – (cid:32) A P A p . s . p P N q | F A “ u{ G p . s . p P N q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 15
Almost flat connections on M . Now we turn to the analytical part of the gaugetheory. For all three-manifolds with boundary, say M for example, we fix a Riemannianmetric on M such that a neighborhood of the boundary is isometric to r , (cid:15) q ˆ B M . Themetric on M induces a metric on manifold with cylindrical ends M defined by (2.7) whichis of the product type on the cylindrical end. When discussing a pair of three-manifolds N ´ and N ` sharing the same boundary, we assume the boundary restrictions of themetrics are isometric. The pair of metrics induce a metric on the closed manifold N defined by (2.9) which is of product type over the neck.The differentiation of gauge fields depends on the choice of a covariant derivative.From now on we fix a smooth flat connection A ref on P Ñ M as a reference connection.By applying gauge transformations, we can arrange that the restriction of A ref to thecylindrical end r , `8q ˆ Σ is equal to d ` B ref where B ref is a flat connection on Q Ñ Σ.Sobolev norms of sections of ad P or ad Q , without further explanation, will be taken withrespect to these reference connections.The following lemma shows that near an almost flat connection on the three-manifoldwith boundary there is always a flat connection. It essentially follows from the transver-sality assumption Hypothesis 2.8 and the implicit function theorem. Lemma 2.13.
Let p ě . There exist (cid:15) “ (cid:15) p ą and C “ C p ą satisfying the followingproperties. Let A be a smooth connection on P Ñ M with } F A } L p p M q ď (cid:15). Then there exists a flat connection A ˚ on P of regularity W ,p satisfying } A ´ A ˚ } W ,p p M q ď C } F A } L p p M q . Proof.
Consider the space W ,p B p M , Λ b ad P q “ ! α P W ,p p M , ad P q | ˚ α | B M “ ) . Consider the linear operator D A : W ,p B p M , Λ b ad P q Ñ L p p M , Λ b ad P q ‘ L p p M , Λ b ad P q defined by D A α “ p˚ d A α, d A ˚ α q . This is Fredholm with index ´ χ p Σ q . We claim thatthere exist (cid:15) “ (cid:15) p ą C “ C p ą } F A } L p p M q ď (cid:15) , D A is surjectiveand there is a right inverse Q A with } Q A } ď C p . Suppose this is not the case, then thereexist a sequence of connections A i with } F A i } L p Ñ D A i is not surjective. Then theweak Uhlenbeck compactness theorem (see [Weh03, Theorem A]) in three dimensionsimplies that a subsequece converges modulo gauge to a flat connection A on M , andthe convergence is weakly in W ,p . This implies that D A i converges to D A in operatornorm. However, by Hypothesis 2.8, D A is surjective as its kernel is the tangent spaceof L M at r A s . This contradiction means as long as } F A } L p is sufficiently small, D A is surjective. Same argument further guarantees the existence of a right inverse withbounded norm as L M is compact. Then one can apply the implicit function theorem (seefor example [MS04, Proposition A.3.4]) to find a nearby flat connection A ˚ satisfying ourrequirement. (cid:3) Remark . Throughout this paper, we adopt the convention that C and (cid:15) representconstants which are allowed to vary from line to line. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 16
The representation variety.
We recall basic facts about the moduli space of flatconnections over a surface, which we often call by the name representation variety. Let Σbe a closed surface and Q Ñ Σ be the nontrivial SO p q -bundle which is nontrivial overeach connected component of Σ. The space A p Q q of smooth connections on Q is an affinespace modelled on Ω p ad Q q . There is a symplectic form defined by ω A p Q q p α, β q “ ż Σ tr p α ^ β q , α, β P Ω p ad Q q . (2.10)The conformal class of the Riemannian metric on Σ defines a compatible almost complexstructure, i.e., J A p Q q α “ ˚ α where the Hodge star operator on 1-forms on Σ only depends on the complex structure. ω A p Q q and J A p Q q make A p Q q an (infinite dimensional) K¨ahler manifold.The space of gauge transformations G p Q q acts on A p Q q through gauge transformations.The action is Hamiltonian, with a moment map µ p B q “ ´ F B P Ω p ad Q q – ´ Lie G p Q q ¯ ˚ The representation variety associated to Q can be identified with the symplectic quotient R Σ – µ ´ p q{ G p Q q . The representation variety R Σ inherits a K¨ahler structure from the symplectic form ω A p Q q and the complex structure J A p Q q (see [Gol84]). The associated K¨ahler metric on R Σ is called the L -metric. It is standard knowledge that the immersion ι : L M í R Σ isLagrangian and totally real with respect to the K¨ahler structure of R Σ .2.5.1. Projection onto the representation variety.
We review the complexification of gaugetransformations on connections on Q (see also discussions in [Fuk98] [Dun13]). Take anopen subset U Ă Σ over which Q can be trivialized. Then we can write B P A p Q q as B “ d ` β where β P Ω p U, so p qq – Ω p U, su p qq . Then B can be viewed as a connectionon a rank two complex vector bundle E Ñ U equipped with a Hermitian metric. A purelyimaginary gauge transformation on Q , written as g “ e i h , h : U Ñ su p q can be viewed as a change of metric on E . Then the Chern connection associated to thisnew metric reads g ˚ B “ d ` e ´ i h B B e i h ´ e ´ i h B B e i h . Here B B ` B B is the covariant derivative on E associated to B . This definition extendsglobally to all sections h of ad Q , and induces another SO p q -connection on Q , denotedby g ˚ B . The infinitesimal version of this action is h ÞÑ ´ ˚ d B h. Following the terminology of Duncan [Dun13, Dun], we define a nonlinear map whichassign to each almost flat connection on the surface to a flat connection via a uniqueimaginary gauge transformation. For p ą (cid:15) ą
0, define A ,p(cid:15) p Q q “ ! B P A ,p p Q q | } F B } L p p Σ q ă (cid:15) ) and A ,p flat p Q q “ ! B P A ,p p Q q | F B “ ) . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 17
Lemma 2.15.
Let p ą . There exist constants (cid:15) “ (cid:15) p ą and C “ C p ą satisfyingthe following conditions. For each B P A ,p(cid:15) p Q q , there exists a unique complex gaugetransformation of the form g “ e i h B where h B P W ,p p Σ , ad Q q such that g ˚ B P A ,p flat p Q q .Moreover, h B P W ,p p Σ , ad p Q qq is a C function of B P A ,p(cid:15) p Q q and there holds theestimate } h B } W ,p p Σ q ď C } F B } L p p Σ q . (2.11) Proof.
We use the implicit function theorem. Consider the map W ,p p Σ , ad Q q Q h ÞÑ F g ˚ B P L p p Σ , ad Q q where g “ e i h . Its linearization at h “ B : “ d ˚ B d B : W ,p p Σ , ad Q q Ñ L p p Σ , ad Q q . This is a Fredholm operator of index zero. If B is flat, then Hypothesis 2.8 implies that∆ B is invertible. Since ∆ B depends on B P A ,p p Q q smoothly, and since R Σ is compact,there is a constant C “ C p ą } ∆ B h } L p p Σ q ě C } h } W ,p p Σ q , @ B P A ,p flat p Q q . Then when (cid:15) is sufficiently small, by Uhlenbeck’s weak compactness, any B P A ,p(cid:15) p Q q issufficiently close to a flat W ,p -connection in the W ,p -norm. It follows that } ∆ B h } L p p Σ q ě C } h } W ,p p Σ q , @ B P A ,p(cid:15) p Q q . Then by applying the implicit function theorem (see for example [MS04, PropositionA.3.4]), one obtains the unique h B satisfying (2.11). The C -dependence of h B on B isalso a consequence of the implicit function theorem. (cid:3) Definition 2.16.
Let p ą (cid:15) p be the one in Lemma 2.15. Define the Narasimhan–Seshadri map NS p : A ,p(cid:15) p p Q q Ñ A ,p flat p Q q (2.12)by NS p p B q “ p e i h B q ˚ B. We know that each flat connection in A ,p flat p Q q is gauge equivalent via a gauge transfor-mation of class W ,p to a smooth flat connection. Then there is a homeomorphism A ,p flat p Q q{ G ,p p Q q – R Σ . The composition of NS p with the projection A ,p flat p Q q Ñ R Σ is denoted by NS p : A ,p(cid:15) p p Q q Ñ R Σ . By abusing names we still call NS p the Narasimhan–Seshadri map. It is well-knownthat the derivative D NS p is complex linear and annihilates infinitesimal complex gaugetransformations. Therefore an ASD instanton over the product S ˆ Q Ñ S ˆ Σ withfibrewise small curvature projects down to a holomorphic curve in the representationvariety via the Narasimhan–Seshadri map. This fact is stated in precise terms as follows.
Proposition 2.17.
Let S Ă C be an open subset and A “ d S ` φds ` ψdt ` B be asmooth connection on S ˆ Q satisfying the first equation of (2.4) , i.e. A s ` ˚ A t “ . Suppose for sufficiently small (cid:15) one has B p z q P A , (cid:15) p Q q for all z P S , then the map z ÞÑ NS p B p z qq defines a holomorphic map u : S Ñ R Σ . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 18
Energy identity for pseudoholomorphic curves in the representation variety.
Weshow that the energy of holomorphic curves in the representation variety can be expressedin terms of the Chern–Simons functional. For simplicity we only discuss a special casebut the general formula can be obtained using the same argument. Let P N ˘ Ñ N ˘ be apair of SO p q -bundles over three-manifolds with boundary which satisfy Hypothesis 2.8and Hypothesis 2.11. Let the domain of the holomorphic curve be the strip S r a,b s : “ r a, b s ˆ r´ , s whose coordinates are p s, t q . Let B ˘ S r a,b s Ă B S r a,b s be the ˘ u : S r a,b s Ñ R Σ be a holomorphic map and γ ˘ : B ˘ S r a,b s Ñ L N ˘ be continuous mapssatisfying u | B ˘ S r a,b s “ ι ˘ ˝ γ ˘ . For each s P r a, b s , one can define a piecewise smooth connection A s P A p . s . p P N q asfollows. First, one can find a smooth path of smooth flat connections B s p t q P A flat p Q q parametrized by t P r´ , s such that r B s p t qs “ u p s, t q . Then there is a unique map ψ s : r´ , s Ñ Ω p ad Q q such that B B s B t ´ d B s p t q ψ s “ . Define A s, “ d t ` ψ s p t q dt ` B s p t q P A p P N | N neck q . On the other hand, the boundary values γ ˘ allows one to find a pair A s, ˘ P A flat p P N | N ˘ q such that r A s, ˘ s “ γ p˘ s q P A p P N ˘ q{ G p P N ˘ q , A s, ˘ | B N ˘ “ B s p˘ q . Then A s, ´ , A s, , A s, ` define a piecewise smooth connection A s P A p . s . p P N q . It is easy toshow that its gauge equivalence class is well-defined.To the strip one can associate a four-manifold with boundary as follows. Define N r a,b s : “ p S r a,b s ˆ Σ q Y pB ` S r a,b s ˆ N ` q Y pB ´ S r a,b s ˆ N ´ q where we glue along the common boundaries B ˘ S r a,b s ˆ Σ. One defines an SO p q -bundle P Ñ N r a,b s as P “ p S r a,b s ˆ Q q Y pB ` S r a,b s ˆ P N ` q Y pB ´ S r a,b s ˆ P N ´ q . The restriction of P to each boundary component of N r a,b s is isomorphic to P N Ñ N .Then one can define the Chern–Simons action CS : ` A p . s . p P N q{ G p . s . p P N q ˘ Ñ R defined by CS pr A a s , r A b sq “ ż N r a,b s tr p F A ^ F A q (2.13)where A is any piecewise smooth connection on N r a,b s extending the boundary values r A a s and r A b s . One has the following formula for holomorphic maps defined over thestrip. Proposition 2.18.
The energy of u is equal to CS pr A a s , r A b sq . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 19
Proof.
For completeness we give a proof. We define a piecewise smooth connection A on N r a,b s as follows. Since u is smooth, one can find a family of smooth connections B p z q P A p Q q parametrized smoothly by z P S r a,b s such that r B p z qs “ u p z q . Since R Σ is a free symplectic quotient, there are unique φ p z q , ψ p z q P Ω p ad Q q such that B s B p z q ´ d B p z q φ p z q P ker d B p z q X ker d ˚ B p z q , B t B p z q ´ d B p z q ψ p z q P ker d B p z q X ker d ˚ B p z q . Then define A “ d S ` φ p z q ds ` ψ p z q dt ` B p z q which is a smooth connection on P restricted to S r a,b s ˆ Σ. Since the map z ÞÑ r B p z qs isholomorphic, one has B s B p z q ´ d B p z q φ p z q ` ˚pB t B p z q ´ d B p z q ψ p z qq “ . Moreover, since the linear map D NS : ker d B p z q X ker d ˚ B p z q Ñ T r B p z qs R Σ is an isometry,we have |B s u | “ |B s B ´ d B φ p z q| “ ż Σ tr ´ pB s B ´ d B φ q ^ pB t B ´ d B ψ q ¯ . (2.14)On the other hand, for each boundary point p s, ˘ q P B ˘ S r a,b s , using the map γ ˘ onecan find a family of flat connections A s, ˘ P A flat p P N q parametrized smoothly by s suchthat r A s, ˘ s “ γ ˘ p s q @p s, ˘ q P B ˘ S r a,b s . Then define A ˘ “ d s ` A s, ˘ P A pB ˘ S r a,b s ˆ P N ˘ q . This is a smooth connection on P restricted to B ˘ S r a,b s ˆ N ˘ . Then A and A ˘ togetherdefine a piecewise smooth connection A on P .We calculate the Chern–Simons action using A . Indeed, since A s, ˘ is flat, one hastr p F A ^ F A q| B ˘ S r a,b s ˆ N ˘ “ . On the other hand, over S r a,b s ˆ Σ one has F A “ F B p z q ` ds ^ pB s B p z q ´ d B φ q ` dt ^ pB t B p z q ´ d B ψ q ` pB s ψ ´ B t φ ` r φ, ψ sq ^ dsdt. Since F B p z q ”
0, one hastr p F A ^ F A q “ tr ´ pB s B ´ d B φ q ^ pB t B p z q ´ d B ψ q ¯ ^ dsdt. By (2.14), we see CS pr A a s , r A b sq “ ż N r a,b s tr p F A ^ F A q “ ż S r a,b s |B s u | dsdt “ E p u q . (cid:3) The rescaled equation and interior compactness
In this section we consider the ASD equation over the product of two surfaces. In theadiabatic limit of the rescaled version, we recall an interior estimate by Dostoglou–Salamon[DS94, DS07] which leads to the compactness modulo bubbling results (Theorem 3.3).We also give a refined version of the interior estimate near the boundary (Corollary 3.6)which will be useful in the next section.
OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 20
An interior estimate for the rescaled equation.
First we recall the notion ofthe rescaled ASD equations. Let ρ be a positive number. Let S Ă C be an open subset.Let ϕ ρ : C Ñ C be the multiplication z ÞÑ ρz . Recall that a connection A on ϕ ρ p S q ˆ Q can be writtenin components as A “ d ϕ ρ p S q ` φds ` ψdt ` B . Denote A “ ϕ ˚ ρ A “ d S ` φ ds ` ψ dt ` B . Recall the notations introduced in (2.2) and (2.3). Then the ASD equation on A can berewritten as the following equation on A A s ` ˚ A t “ , ˚ κ A ` ρ µ A “ . (3.1)We call this equation the ρ -ASD equation . Define the rescaled energy density function e ρ p z q : “ } A s } L pt z uˆ Σ q ` ρ } µ A } L pt z uˆ Σ q (3.2)and the rescaled energy E ρ p A ; S q : “ ż S e ρ p z q dsdt. The basic relation between the rescaled energy density and the original energy density is e ρ p z q “ ρ } F A } L pt ρz uˆ Σ q , @ z P S. Therefore we have E p A ; ϕ ρ p S qq “ } F A } L p ϕ ρ p S qˆ Σ q “ E ρ p A ; S q . When the set S is understood from the context, we abbreviate E ρ p A ; S q by E ρ p A q . Wealso introduce another density in terms of fibrewise L -norm: e ρ p z q : “ } A s } L pt z uˆ Σ q ` ρ } µ A } L pt z uˆ Σ q . In [DS94, DS07] Dostoglou–Salamon obtain several important estimates about therescaled equation. These estimates are essential in proving the convergence of rescaledinstantons towards holomorphic curves. Here we recall one of their estimates as follows.
Theorem 3.1. [DS07, Corollary 1.1]
Let S Ă C be an open set, K Ă S be a compactsubset, and c ą . Then there exist constants C ą and T ą such that the followingholds. Let A “ p B , φ , ψ q be a solution to (3.1) over S ˆ Σ with ρ ě T satisfying sup z P S e ρ p z q “ sup z P S ´ } A s } L p Σ q ` ρ } µ A } L p Σ q ¯ ď c . (3.3) Then there holds } A s } L p K ˆ Σ q ` ρ } µ A } L p K ˆ Σ q ď C b E ρ p A ; S q . Remark . Dostoglou–Salamon’s estimates are valid for the case that the ρ -ASD equationhas a type of holonomic perturbation term. Such a perturbation induces a family ofHamiltonian function on R Σ . See [DS94, Section 7] for more details. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 21
Adiabatic limit.
Now we consider the adiabatic limit of the rescaled ASD equationtowards holomorphic maps in the representation variety. Let S Ă C be an open subsetand S i Ă S be an exhausting sequence of open subsets. The conclusions of the followingtheorem is well-known and is essentially a consequence of Dostoglou–Salamon’s estimatesin [DS94, Section 7] and [DS07]. Theorem 3.3. (cf. [DS94, Theorem 9.1] [Dun, Lemma 3.5] [Nis10, Theorem 1.2] )Let ρ i Ñ `8 be a sequence of positive numbers diverging to infinity. Let S Ă C be an open subset and S i Ă S be an exhausting sequence of open subsets. Let A i “ d S ` φ i ds ` ψ i dt ` B i p z q be a sequence of solutions to the ρ i -ASD equation over S i ˆ Σ .Let e ρ i : S i Ñ r , `8q the rescaled energy density function of A i . Suppose lim sup i Ñ8 E ρ i p A i q ă 8 . (3.4) Then there exists a subsequence (still indexed by i ) and a holomorphic map with mass p u , W , m q : S Ñ R Σ satisfying the following condition.(a) For every point z P S there holds lim r Ñ lim i Ñ8 E ρ i p A i ; B r p z q ˆ Σ q “ m p z q . (b) For any precompact open subset K Ă S (cid:114) W , for i sufficiently large, B i p z q iscontained in the domain of the Narasimhan–Seshadri map NS for all z P K ,hence A i induces a holomorphic map u i : K Ñ R Σ , u i p z q : “ NS p B i p z qq . Moreover, the sequence of maps u i converges to u in C p K q and the energydensity e ρ i converges to the energy density of u in C p K q .(c) For any compact subset K Ă S containing W , there holds lim i Ñ8 E ρ i p A i ; K ˆ Σ q “ E p u ; K q ` ż K m . In particular, W “ H if and only if for all compact K Ă S there holds lim sup i Ñ8 } e ρ i } L p K q ă 8 . Definition 3.4.
Let S Ă C be an open subset and S i Ă S be an exhaustive sequence ofopen subsets. Let ρ i Ñ `8 be a sequence of positive numbers. Let A i be a sequence ofsolutions to the ASD equation over ϕ ρ i p S i qˆ Σ. Let ˜ u “ p u , W , m q be a holomorphicmap from S to R Σ with mass. We say that A i converges to ˜ u along t ρ i u if for thecorresponding sequence of solutions A i of the ρ i -ASD equation over S i ˆ Σ, conditions(a), (b), and (c) of Theorem 3.3 are satisfied.3.3.
Estimates near the boundary.
In this section we extend certain estimates ofDostoglou–Salamon near the boundary of the domain. Such an extension will be useful inthe next section. Let B R “ B R p q Ă C be the radius R open disk centered at the origin. Lemma 3.5.
There exist (cid:15) ą , T ą , and C ą satisfying the following conditions.Let A “ d C ` φds ` ψdt ` B p z q be a solution to the ρ -ASD equation over B ˆ Σ satisfying ρ ě T , E ρ p A q ě (cid:15) and sup B e ρ ď ρ . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 22
Then for all r P p , q there holds sup B r e ρ ď C p ´ r q . (3.5) Proof.
Suppose this is not the case. Then there exist a sequence ρ i Ñ 8 , a sequence ofsolutions A i to the ρ i -ASD equation over B ˆ Σ, and a sequence of points z i P B suchthat E ρ i p A i q Ñ
0, sup e ρ i ď ρ i , andlim i Ñ8 p ´ r i q e ρ i p z i q “ 8 , where r i “ | z i | . (3.6)Then it follows that ρ i ě e ρ i p z i q " p ´ r i q . Here a i " b i (equivalently b i ! a i ) means a i { b i Ñ `8 . Denote ˜ ρ i : “ ρ i p ´ r i q whichdiverges to infinity. Then define the rescaling ϕ i : B Ñ B ´ r i p z i q , w ÞÑ z i ` p ´ r i q w. The sequence A i is pulled back to a sequence of solutions ˜ A i to the ˜ ρ i -ASD equation over B ˆ Σ whose energy converges to zero. On the other hand, (3.6) implies that e ˜ ρ i p q Ñ `8 . (3.7)By Theorem 3.3, this sequence converges to the constant holomorphic map, implying that e ˜ ρ i converges to zero uniformly over compact subsets of B . This contradicts the interiorestimate of Theorem 3.1 applied to ˜ A i and (3.7). (cid:3) There is also a more refined case of the inequality (3.1) of Theorem 3.1 which will beuseful in the next section. Denote the open half disk in the upper half plane by B ` r : “ B ` r p q : “ t z P C | Im z ě , | z | ă r u . Corollary 3.6.
There exist (cid:15) ą , C ą , T ą satisfying the following conditions.Suppose A “ d C ` φds ` ψdt ` B p z q is a solution to the ρ -ASD equation over Int B ` ˆ Σ for ρ ě T satisfying E ρ p A q ď (cid:15), sup z P Int B ` } F B p z q } L p Σ q ď (cid:15), sup B ` e ρ ď ρ . Then for all z “ s ` i t P Int B ` there holds } A s } L pt z uˆ Σ q ` ρ } µ A } L pt z uˆ Σ q ď Ct b E ρ p A ; B t p z q ˆ Σ q . (3.8) Proof.
The proof is essentially from the proof of [DS94, Theorem 7.1]. Define two functions h , h : Int B ` Ñ r , `8q by h p z q “ ´ } A s } L pt z uˆ Σ q ` ρ } F B p z q } L p Σ q ¯ “ e ρ p z q ,h p z q “ ´ ρ } d B p z q A s } L p Σ q ` ρ } d B p z q ˚ A s } L p Σ q ` } ∇ s A s } L p Σ q ` } ∇ t A s } L p Σ q ` } d B p z q κ A } L p Σ q ` ρ ´ } ∇ s κ A } L p Σ q ` ρ ´ } ∇ t κ A } L p Σ q ¯ . Then by the explicit calculation in [DS94, p. 616], one obtains∆ h “ h ` x A s , ˚r A s ^ κ A sy ě } d B p z q κ A } L p Σ q ` x A s , ˚r A s ^ κ A sy . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 23
When (cid:15) is small enough, the condition } F B p z q } L p Σ q ď (cid:15) implies that B p z q is sufficientlyclose to a flat connection on Q Ñ Σ in the L -metric. Since there is no reducible flatconnections on Q and R Σ is compact, there is a constant a ą } d B p z q κ A } L p Σ q ě a } κ A } L p Σ q . On the other hand, by Lemma 3.5, when ρ is sufficiently large and the total energy of A is sufficiently small, for some C ą } A s } L pt z uˆ Σ q ď Ct ´ , @ z “ s ` i t P Int B ` . Hence ˇˇ x A s , ˚r A s ^ κ A sy ˇˇ ď Cat } d B p z q κ A } L p Σ q } A s } L p Σ q . Therefore, for a suitably modified value of C there holds∆ h ě ´ C t } A s } L p Σ q ě ´ Ct h . Then by Lemma 3.7 below, one has h p z q ď Ct ż B t p z q h “ Ct E ρ p A ; B t p z q ˆ Σ q . By the definition of h , (3.8) follows. (cid:3) The following mean value estimate was used in the above proof.
Lemma 3.7. [DS94, Lemma 7.3]
Let u : B r Ñ r , `8q be a C -function satisfying ∆ u ě ´ au, where a ě . Then there holds u p q ď π ´ a ` r ¯ ż B r u. The isoperimetric inequality
It is not too far away from results of the previous section to the compactification ofthe moduli space of instantons over the product of the complex plane and the compactsurface, namely Theorem 1.5. For the other type of noncompact four-manifold, namelythe product R ˆ M where M is the three-manifold with cylindrical end, there is anotherdifficulty. As approaching to the infinity of the R -direction, the ASD equation over thecompact part R ˆ M is almost like a Lagrangian boundary condition for the part ASDequation over H ˆ Σ. In the theory of holomorphic curves, the Lagrangian boundarycondition allows one to extend interior elliptic estimates to the boundary which leadsto compactness near the boundary. However, in our situation, the failure of being anactual Lagrangian boundary condition prevents one to have elliptic estimates near theboundary, at least not in a straightforward way. Indeed, one should view the “seam” R ˆ Σ between R ˆ M and H ˆ Σ as giving a Lagrangian seam condition given by aninfinite dimensional Lagrangian correspondence (see [WW15]). One possible approach ofour compactness problem would be based on certain hard estimates as did in [BW18] forfinite dimensional holomorphic quilts (see another approach in [Dun, Section 4]).In this paper, instead, we take a less analytic approach. The main idea is to viewthe ASD equation over R ˆ M as a gradient line of the Chern–Simons functional on theclosed three-manifold M double with respect to a time-dependent metric. The asymptoticbehavior as well as the compactness problem over the part R ˆ M can both be treatedvia an isoperimetric inequality. Roughly speaking, for a closed Riemannian three-manifold OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 24 N , if the Chern–Simons functional for connections on an SO p q -bundle P Ñ N is Morseor Morse–Bott, then for an almost flat connection A P A p P q , one can define a “localChern–Simons action,” which is the Chern–Simons action of A relative to a certain nearbyflat connection. The isoperimetric inequality says that the local action can be controlledby } F A } L p N q . This is analogous to the isoperimetric inequality in symplectic geometry(see [MS04, Section 4.4] [Po´z94, Chapter 3]).In this section we prove the isoperimetric inequality and certain monotonicity propertiesof solutions to the ASD equation (which we call the annulus lemma). We will also derivea diameter estimate which will be needed for the compactness theorem as well as theasymptotic behavior and energy quantization property of instantons over R ˆ M .4.1. The isoperimetric inequality.
In this subsection we derive the isoperimetricinequality. Let N ´ , N ` be three-manifold with boundary with diffeomorphic boundaryand P N ˘ Ñ N ˘ be SO p q -bundles with isomorphic boundary restrictions, all of whichsatisfy Hypothesis 2.8 and Hypothesis 2.11. Let N be the closed three-manifold obtainedby gluing N ´ , p N ` q op and a neck region N neck “ r´ , s ˆ Σ and P N Ñ N be the glued SO p q -bundle. For any piecewise smooth connection A P A p . s . p P N q , if its restriction tothe neck region is A | r´ , sˆ Σ “ d t ` η p t q dt ` B p t q where d t is the exterior differentiation in the coordinate t P r´ , s , we define the L -lengthof A to be l p A q : “ ż ´ } B p t q ´ d B p t q η p t q} L p Σ q dt. Lemma 4.1.
Given p ě , there exist (cid:15) ą and C ą satisfying the following condition.Let A be a piecewise smooth connection on P N whose restriction to N neck “ r´ , s ˆ Σ is A | r´ , sˆ Σ “ d t ` η p t q dt ` B p t q . Suppose } F A } L p p N ´ Y N ` q ` l p A q ď (cid:15). (4.1) Then there exists a piecewise smooth flat connection A ˚ P A p . s . flat p P N q such that } A ´ A ˚ } W ,p p N ´ Y N ` q ď C ` } F A } L p p N q ` l p A q ˘ . (4.2) Moreover, A ˚ is gauge equivalent to another connection A P A p . s . flat p P N q satisfying(a) The restriction of A to N neck is d t ` η p t q dt ` B p t q .(b) There holds sup t Pr´ , s } B p t q ´ B p t q} L p Σ q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ (4.3) and } A ´ A } L p N ´ Y N ` q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . (4.4) Proof.
First, notice that the statement of this lemma is gauge invariant. Hence we mayassume that η p t q ” dt component. Then for any pair a, b P r´ , s there holds } B p b q ´ B p a q} L p Σ q “ ż ba } B p t q} L p Σ q dt ď l p A q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 25
Let C ą A but may vary in the context. Let (cid:15) besmaller than the (cid:15) p of Lemma 2.13. Then by Lemma 2.13 there exist flat W ,p -connections A ˘ on N ˘ such that } A ´ A ˘ } W ,p p N ˘ q ď C } F A } L p p N ˘ q . We can replace A ˘ by nearby smooth connections such that this bound is still valid for aslightly larger C . Let B ˘ be the boundary restrictions of A ˘ over B N ˘ – Σ. Then } B ˘ ´ B p˘ q} L p Σ q ď C } B ˘ ´ B p˘ q} L p p Σ q ď C } A ´ A ˘ } W ,p p N ˘ q ď C } F A } L p p N ˘ q . On the other hand, one has } B ` ´ B ´ } L p Σ q ď } B ` ´ B p q} L p Σ q ` } B p q ´ B p´ q} L p Σ q ` } B p´ q ´ B ´ } L p Σ q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . (4.5)Let d R Σ be the distance function on R Σ induced from the L -metric. The estimate(4.5) implies that for some C ą d R Σ ´ pr B ´ s , r B ` sq , ∆ R Σ ¯ ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . The immersions ι ˘ : L N ˘ í R Σ pulls back a metric which induces a distance function d L N ˘ on L N ˘ . Let d L N ´ ˆ L N ` be the product metric on L N ´ ˆ L N ` . Since ι ´ and ι ` intersect cleanly, there holds d L N ´ ˆ L N ` ´ pr A ´ s , r A ` sq , p ι ´ , ι ` q ´ p ∆ R Σ q ¯ ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . Moreover, the topology on L N ˘ induced by d L N ˘ is the same as the topology inducedfrom the Banach space topology on A ,p p P N ˘ q . Hence one can find a pair of smooth flatconnections A ˚˘ on N ˘ such that } A ˘ ´ A ˚˘ } W ,p p N ˘ q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ (4.6)and such that r A ˚´ | B N ´ s “ r A ˚` | B N ` s P R Σ . Denote B ˚˘ : “ A ˚˘ | B N ˘ . Then by (4.5) and (4.6) one also has } B ˚´ ´ B ˚` } L p Σ q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . (4.7)Since B ˚´ and B ˚` are gauge equivalent and the group of gauge transformations G p Q q is connected (since we only take SU p q -valued gauge transformations), there is a gaugetransformation g relating B `´ and B ˚` which can be homotopied to the identity. Sucha homotopy can be used to extend A ˚´ and A ˚` to a piecewise smooth flat connection A ˚ P A p . s . flat p P N q which satisfies (4.2).Now we would like to modify A ˚ via a suitable gauge transformation so that (4.3) and(4.4) holds. Indeed, the gauge transformation g can be written as g “ e h and (4.7) impliesthat } h } W , p Σ q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . (4.8)Recall that for any smooth manifold Y with boundary B Y , the boundary restrictiondefines a trace operator H s p Y q Ñ H s ´ pB Y q which admits a bounded right inverse. Therefore, by the bound (4.8) there exists a smoothextension of h to N ` , denoted by h ` such that } h ` } H p N ` q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 26
Then we can define a piecewise smooth flat connection A “ $’&’% A ˚´ , N ´ ,d ` B ˚´ , r´ , s ˆ Σ , p e h ` q ˚ A ˚` , N ` Then by the Sobolev embedding W , p N ´ q Ñ L p N ´ q and the Sobolev embedding (forfractional Sobolev norms, see [NPV, Theorem 6.5]) H p N ` q “ W , p N ` q Ñ L p N ` q ,there holds } A ´ A } L p N ´ Y N ` q ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ . The condition (4.3) is also easy to verify. (cid:3)
The above lemma allows one to define a local Chern–Simons action.
Definition 4.2.
Let A be a piecewise smooth connection on P N satisfying (4.1). Thenthere exists a nearby flat connection A ˚ on P N satisfying properties of Lemma 4.1. Thenwe define F loc p A q “ CS pr A ˚ s , r A sq where CS pr A ˚ s , r A sq is the Chern–Simons action of r A s relative to r A ˚ s .It is easy to verify that the value of the local Chern–Simons functional is independentof the choice of the reference flat connection as long as the reference connection belongsto the same connected component L N . We denote F loc p A q “ F Z p A q where Z Ă L N is the connected component of the nearby flat connection r A ˚ s .Lemma 4.1 allows us to derive the following isoperimetric inequality. The proof ofthis inequality (for the SU p q case) has appeared in [Flo88a] for the case that L N isa transverse intersection (see also [Don02, Proof of Theorem 4.2]) and in the proof of[Fuk96, Lemma 7.13] for the case that L N is a clean intersection. Theorem 4.3 (Isoperimetric Inequality) . Given p ě , there exist (cid:15) ą and C ą suchfor all piecewise smooth connection A on P N satisfying (4.1) , there holds | F loc p A q| ď C ´ } F A } L p p N ´ Y N ` q ` l p A q ¯ . (4.9) Proof.
By Lemma 4.1, there is a nearby flat connection A and the local action is definedby Definition 4.2. Then by Lemma 4.1, (4.4), and formula (2.1), one has ż N ˘ tr „ d A p A ´ A q ^ p A ´ A q ` p A ´ A q ^ p A ´ A q ^ p A ´ A q ď C } F A } L p N ˘ q } A ´ A } L p N ˘ q ` C } A ´ A } L p N ˘ q ď C ´ } F A } L p p N ´ Y N ` q ` l p A q ¯ . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 27
Using the fact that A ´ A | N neck “ B p t q ´ B and (4.3), one has ż N neck tr „ d A p A ´ A q ^ p A ´ A q ` p A ´ A q ^ p A ´ A q ^ p A ´ A q “ ż r´ , sˆ Σ tr „ d A p A ´ A q ^ p A ´ A q ď C ż r´ , sˆ Σ | B p t q|| B p t q ´ B | dtd vol Σ ď C ż ´ } B p t q} L p Σ q } B p t q ´ B } L p Σ q dt ď C ` } F A } L p p N ´ Y N ` q ` l p A q ˘ ż ´ } B p t q} L p Σ q dt ď C ´ } F A } L p p N ´ Y N ` q ` l p A q ¯ . Then (4.9) follows. (cid:3)
In practice it is more convenient to use the following version of the isoperimetricinequality. Notice that the L -length l p A q can be bounded by } F A } L p N neck q . So the p “ L N is aclean intersection, the following version follows from an estimate of the Hessian of theChern–Simons functional (see[Fuk96]). Theorem 4.4 (Isoperimetric Inequality) . There exist (cid:15) ą and c P ą satisfying thefollowing condition. Let A be a piecewise smooth connection on P N Ñ N satisfying } F A } L p N q ď (cid:15). Then the local Chern–Simons action F loc p A q is well-defined and there holds | F loc p A q| ď c P } F A } L p N q The annulus lemma.
Now we turn to the decay property of the energy for ASDinstantons. For 0 ă r ă R ă 8 define the open annulus and the half annulus by Ann p r, R q : “ t z P C | r ă | z | ă R u , Ann ` p r, R q “ Ann p r, R q X H . The half annulus has two boundary components B ˘ Ann ` p r, R q : “ t s P B H – R | r ă ˘ s ă R u . Define a noncompact four-manifold N r,R as N r,R “ p Ann ` p r, R q ˆ Σ q Y pB ` Ann ` p r, R q ˆ N ` q Y pB ´ Ann ` p r, R q ˆ N ´ q . Equip N r,R with the product metric. The bundle P N Ñ N also extends to a bundle P Ñ N r,R .It is convenient to use the polar coordinates over the neck region. We identify the neckregion with N neck – r´ , s ˆ Σ – r , π s ˆ Σwith the coordinate t P r´ , s corresponding to θ : “ π p t ` q P r , π s . Suppose A is asmooth connection on N r,R . One can identify A with a piecewise smooth connectionon the product p log r, log R q ˆ N as follows. Over p log r, log R q ˆ N ˘ , define A as thepullback of A via the diffeomorphism p log r, log R q ˆ N ˘ – B ˘ Ann ` p r, R q ˆ N ˘ , p τ, x q ÞÑ p e τ , x q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 28
Over p log r, log R q ˆ r , π s ˆ Σ, define A as the pullback of A via the diffeomorphism p log r, log R q ˆ r , π s ˆ Σ Ñ Ann ` p r, R q ˆ Σ , p τ, θ, w q ÞÑ p e τ ` i θ , w q . Then A is a piecewise smooth connection, giving a family of piecewise smooth connections A e τ on the bundle P N Ñ N parametrized by ρ “ e τ P p r, R q . Then we can write A ρ | r ,π sˆ Σ “ d θ ` η ρ p θ q dθ ` B ρ p θ q . It is easy to see that if A is a solution to the ASD equation over N r,R , then E p A ; N r,R q“ ż log R log r ˆ e τ } F A eτ } L p N ´ Y N ` q ` } B B e τ B θ ´ d B eτ η e τ } L p N neck q ` e τ } F B eτ } L p N neck q ˙ dτ. Then one has E p A ; N r,R q ě ż log R log r } F A eτ } L p N q dτ, @ R ą r ě . (4.10)Moreover, denote the slice of N r,R at radius ρ P p r, R q by N ρ : “ N ´ Y pr , ρπ s ˆ Σ q ˆ N ` Ă N r,R , which is diffeomorphic to N with a different neck length. One can see that } F A ρ } L p N q ď ρ } F A ρ } L p N ρ q , @ ρ ě . (4.11)Now we state and prove the annulus lemma. Define the exponential factor δ P : “ c ´ P where c P is the isoperimetric constant of Theorem 4.4. The constant depends on thebundle P N Ñ N and the Riemannian metric on N . But by abuse of notation we do notdistinguish them since we only consider finitely many examples of such a bundle P N Ñ N . Proposition 4.5 (Annulus Lemma) . There exist (cid:15) ą , C ą satisfying the followingconditions. Let A be a solution to the ASD equation over N r,R with r ě and log R ´ log r ě . Assume E p A q ă (cid:15). Then for ď s ď p log R ´ log r q there holds E p A ; N e s r,e ´ s R q ď Ce ´ δ P s E p A ; N r,R q . (4.12) Proof.
One can identify A with a piecewise smooth connection over p r, R qˆ N as explainedabove. In particular, one obtains a family of connections A ρ on P N Ñ N parametrizedby ρ P p r, R q . Define I (cid:15) p r, R q : “ t ρ P p r, R q | } F A ρ } L p N q ă (cid:15) u . By (4.10) one has (cid:15) ą E p A q ě ż log R log r } F A eτ } L p N q dτ. Therefore in every subinterval p a, a ` q Ă p log r, log R q of length one there exists a point τ P p a, a ` q with e τ P I (cid:15) p r, R q . Then by Lemma 4.1 and Definition 4.2, the local actionof A e τ , denoted temporarily by F p A e τ q , can be defined for e τ P I (cid:15) p r, R q .We would like to show that these local actions, which a priori are not defined for all τ ,extend to a smooth function in τ . In fact there is a map I (cid:15) p r, R q Ñ π p L N q “ : t Z , . . . , Z m u OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 29 such that F p A ρ q “ F Z ρ p A ρ q , @ ρ P I (cid:15) p r, R q where Z ρ is the image of ρ and F Z ρ p A ρ q is the Chern–Simons action of A ρ relative to theconnected component Z ρ . We claim that F Z ρ p A ρ q “ F Z ρ p A ρ q , @ ρ, ρ P I (cid:15) p r, R q . (4.13)Indeed, by Lemma 2.1 we know that (suppose ρ ă ρ ) | F Z ρ p A ρ q ´ F Z ρ p A ρ q| “ E p A ; N ρ,ρ q ă (cid:15). Moreover, by the isoperimetric inequality (Theorem 4.4) we know that | F Z ρ p A ρ q ´ F Z ρ p A ρ q| ď c P ´ } F A ρ } L p N q ` } F A ρ } L p N q ¯ ď C(cid:15).
Therefore, when (cid:15) is small enough, the difference between F Z ρ p A ρ q and F Z ρ p A ρ q is smallerthan the minimal difference between critical values of the Chern–Simons functional on N . Hence (4.13) is true. Then we can fix a connected component Z “ Z ρ for some ρ P I (cid:15) p r, R q and define F p ρ q : “ F Z p A ρ q , @ ρ P p r, R q . This is a smooth non-increasing function and F p ρ q agrees with the local action of A ρ when ρ P I (cid:15) p r, R q . Then for s P r , p log R ´ log r qs , define J p s q : “ F p re s q ´ F p Re ´ s q “ ż Re ´ s re s } F A } L p N ρ q dρ. We would like to derive a certain differential inequality of J p s q . By (4.11) and the factthat Re ´ s ě re s ě J p s q “ ´ re s } F A } L p N res q ´ Re ´ s } F A } L p N Re ´ s q ď ´} F A res } L p N q ´ } F A Re ´ s } L p N q . If both re s and Re ´ s are in I (cid:15) p r, R q , then by the isoperimetric inequality, one has J p s q ď ´} F A res } L p N q ´ } F A Re ´ s } L p N q ď ´ δ P ` F p re s q ´ F p Re ´ s q ˘ “ ´ δ P J p s q . If re s and/or Re ´ s are not in I (cid:15) p r, R q , i.e., when } F A res } L p N q ą (cid:15) and { or } F A Re ´ s } L p N q ą (cid:15), there exists s and/or s P r s ´ , s s Ă r , p log R ´ log r qs such that } F A res } L p N q ď (cid:15) and { or } F A Re ´ s } L p N q ď (cid:15). Then by the isoperimetric inequality and the monotonicity of F , one obtains J p s q ď ´} F A res } L p N q ´ } F A Re ´ s } L p N q ď ´} F A res } L p N q ´ } F A Re ´ s } L p N q ď ´ δ P p F p re s q ´ F p Re ´ s qq ď ´ δ P p F p re s q ´ F p Re ´ s qq “ ´ δ P J p s q . It follows that J p s q decays exponentially as s increases and hence (4.12) is proved. (cid:3) The above annulus lemma contains two special cases corresponding to the two specialcases of Remark 2.12. In the first case when N ´ – p N ` q op – M , the four-manifold N r,R is an open subset of R ˆ M . In this case we denote M r,R : “ N r,R , where N – M double . In the second case when N ´ – p N ` q op – r , π s ˆ Σ (whose boundary is two copies ofΣ instead of one), one has a diffeomorphism N – S ˆ Σ. Although the four-manifold N r,R is not isometric to Ann p r, R q ˆ Σ there is a constant a ą r OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 30 and R such that N r,R is an open subset of Ann p r ´ a, R ` a q ˆ Σ. Then one obtains acorresponding version of annulus lemma for instantons over
Ann p r, R q ˆ Σ. Corollary 4.6.
There exist (cid:15) ą , C ą , and r ą satisfying the following conditions.Let A be a solution to the ASD equation over Ann p r, R qˆ Σ with r ě r and log R ´ log r ě and assume E p A q ă (cid:15) . Then for ď s ď p log R ´ log r q there holds E p A ; Ann p e s r, e ´ s R q ˆ Σ q ď Ce ´ δ P s E p A ; Ann p r, R q ˆ Σ q . In both of the two special cases, we would like to extend the annulus lemma to allow r “
0. For R ą
0, define M R : “ p B ` R ˆ Σ q Y pp´ R, R q ˆ M q where we glue the common boundary p´ R, R q ˆ
Σ in the obvious manner. The four-manifold can be formally viewed as the annulus M ,R considered above. Proposition 4.7.
There exist (cid:15) ą , C ą satisfying the following conditions. Let A bea solution to the ASD equation over M R with log R ě and E p A q ă (cid:15) . Then for s ě there holds E p A ; M Re ´ s q ď Ce ´ δ P s E p A ; M R q . (4.14) Proof.
Similar to the proof of Proposition 4.5, for all ρ P r , R s , one can define a relativeChern–Simons action F p A ρ q which agrees with the local action whenever } F A ρ } L p N q ď (cid:15) .We claim that when Re ´ s ě E p A ; M Re ´ s q “ ´ F p Re ´ s q . (4.15)Notice that the connection A : “ A | t uˆ M defines a not necessarily flat connection A double0 over M double by doubling A and by Lemma 2.1 one has E p A ; M Re ´ s q “ CS pr A double0 s , r A Re ´ s sq . On the other hand, by Uhlenbeck compactness, if (cid:15) is sufficiently small, the restriction of A | M is sufficiently close to a flat connection. Moreover, for every ρ P r , s , } F A ρ } L p N q is sufficiently small. Hence there exists a nearby flat connection A ˚ ρ P A p . s . flat p P N q such that F p ρ q “ F loc p A ρ q “ CS pr A ˚ ρ s , r A ρ sq . The fact that A | M is close to a flat connection implies that } A ρ | N ´ ´ A ρ | N ` } W , p M q isvery small. Therefore, the gauge equivalence class of the nearby flat connection A ˚ ρ mustbe in ∆ L M Ă L M ˆ L M , but not a double point in R L M . Hence A ˚ ρ is gauge equivalent tothe double of a flat connection on M . Therefore CS pr A double0 s , r A ˚ ρ sq “ E p A ; M Re ´ s q “ CS pr A double0 s , r A Re ´ s sq “ CS pr A ˚ ρ s , r A Re ´ s sq “ ´ F p Re ´ s q . Abbreviate J p s q “ ´ F p Re ´ s q . By (4.15), (4.11), and the condition Re ´ s ě J p s q “ ´ Re ´ s } F A } L p N Re ´ s q ď ´} F A Re ´ s } L p N q . Assume s ě
1. If } F A Re ´ s } L p N q ď (cid:15) , then by the isoperimetric inequality one has J p s q ď ´ δ P J p s q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 31 If } F A Re ´ s } L p N q ą (cid:15) , then one can find s P p s ´ , s q Ă r , log R s such that } F A Re ´ s } L p N q ă (cid:15) . Then J p s q ď ´} F A Re ´ s } L p N q ď ´} F A Re ´ s } L p N q ď ´ δ P J p s q ď ´ δ P J p s q . This shows that J p s q decays exponentially for s P r , log R s . So for some C ą E p A ; M Re ´ s q ď Ce ´ δ P s E p A ; M R q , @ s P r , log R s . (4.16)In particular, E p A ; M q ď CR ´ δ P E p A ; M R q . Then by the standard interior estimate for the ASD equation, one has } F A } L p M q ď C a E p A ; M q ď CR ´ δP a E p A ; M R q . On the other hand, for r ď
1, one has Volume p M r q ď Cr . Hence for s ě log R , r “ Re ´ s ď E p A ; M r q ď CrR ´ δ P E p A ; M R q ď Ce ´ s R ´ δ P E p A ; M R q ď Ce ´ δ P s E p A ; M R q . (4.17)Combining (4.16) and (4.17) we obtain (4.14). (cid:3) Similarly one has the following monotonicity property of ASD equation over the productof a disk with the compact surface, although it can be proved by using the mean valueestimate instead of the isoperimetric inequality.
Proposition 4.8.
There exist (cid:15) ą , C ą satisfying the following conditions. Let A be a solution to the ASD equation over B R ˆ Σ with log R ě and E p A q ă (cid:15) . Then for s ě there holds E p A ; B Re ´ s ˆ Σ q ď Ce ´ δ P s E p A ; B R ˆ Σ q . Diameter bound.
To ensure the convergence towards holomorphic curves on theboundary, we need a further diameter control. We first define the notion of diameter.Let S Ă H be an open subset and let A be a solution to the ASD equation over S ˆ Σidentified with a triple p B, φ, ψ q . Suppose for each z P S the connection B p z q P A p Q q iscontained in the domain of the Narasimhan–Seshadri map NS . Then A projects to acontinuous map u : S Ñ R Σ . We definediam p A ; S ˆ Σ q : “ diam p u p S qq : “ sup p,q P S d p u p p q , u p q qq . Clearly this notion is gauge invariant. Moreover, denote M S : “ p S ˆ Σ q Y pB S ˆ M q where the two parts are glued along the common boundary B S ˆ Σ. If A is a solution to M S whose restriction to S ˆ Σ is d S ` φds ` ψdt ` B p z q such that all B p z q is containedin the domain of the Narasimhan–Seshadri map NS , then we definediam p A ; M S q : “ diam p A ; S ˆ Σ q . By using Theorem 3.1, one has the following interior diameter estimate.
Lemma 4.9 (Interior diameter bound) . There exist C ą , T ą , and (cid:15) ą such thatfor any solution to the ASD equation over B R ˆ Σ with R ě T and E p A q ď (cid:15), sup B R } F B p z q } L p Σ q ď (cid:15) there holds diam p A ; B R ˆ Σ q ď C a E p A ; B R ˆ Σ q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 32
Next, we prove the following diameter estimate near the boundary.
Lemma 4.10 (Boundary diameter bound) . There exist C ą , T ą , and (cid:15) ą suchthat, for any solution to the ASD equation A over M R “ M B R with E p A q ď (cid:15) and R ě T satisfying } F A } L p M R q ď , sup B ` R } F B p z q } L p Σ q ď (cid:15) (4.18) there holds diam p A ; M R q ď C a E p A ; M R q . Proof.
Let R and A satisfy the assumption of this lemma with (cid:15) and T undetermined.Suppose the restriction of A to B ` R ˆ Σ by d C ` φds ` ψdt ` B p z q . Denote by u : B ` R Ñ R Σ the holomorphic map defined by z ÞÑ NS p B p z qq . Define the segment Z R : “ (cid:32) z “ s ` i t | ´ R ď s ď R, t “ R ( . Since Z R can be covered by a fixed number of radius R disks contained in B ` R , by Lemma4.9, for T large and (cid:15) small, one hasdiam p A ; Z R ˆ Σ q ď C a E p A ; M R q . (4.19)Hence it remains to show thatsup ´ R ď s ď R sup ď t ,t ď R dist p u p s ` i t q , u p s ` i t qq ď C a E p A ; M R q . (4.20)To estimate the distance between u p s ` i t q and u p s ` i t q for 0 ď t , t ď R , weconsider the rescaled equation and use the estimate of Corollary 3.6. The restriction of A to B ` R ˆ Σ can be pulled back to a solution A “ d C ` φ ds ` ψ dt ` B p z q to the R -ASDequation over B ` ˆ Σ satisfying E R p A q ď (cid:15) andsup B ` e R ď R , sup B ` } F B p z q } L p Σ q ď (cid:15). Then by Corollary 3.6, when T is sufficiently large and (cid:15) is sufficiently small one has } A s } L pt s ` i t uˆ Σ q ď Ct a E R p A ; B t p z q ˆ Σ q , @| s | ď , ă t ď . Moreover, by Proposition 4.7 for the case N “ M double , for some constant C ą A and R one has E R p A ; B t p z qˆ Σ q ď E R p A ; B ` t p s qˆ Σ q “ E p A ; M B ` Rt p s q q ď Ct δ P ¨ E p A ; M R q , @ t P p , s . Therefore one has } A s } L pt s ` i t uˆ Σ q ď Ct ´ ` δP a E p A ; M R q , @ t P p , s . Now for z “ s ` i t and z “ s ` i t with | s | ď ă t ď t ď u p z q and u p z q . Using a suitable gauge transformation wecan assume ψ p s ` i t q “ t P r , s . Then } B p z q ´ B p z q} L p Σ q ď ż }B τ B p s ` i τ q} L p Σ q dτ “ ż } A t } L p Σ q dτ “ ż } A s } L p Σ q dτ ď C a E p A ; M R q ż τ ´ ` δP dt ď C a E p A ; M R q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 33
Then by Lemma 2.15, for | s | ď t , t P p , s , one hasdist p u p Rs ` i Rt q , u p Rs ` i Rt qq ď } NS p B p z qq ´ NS p B p z qq} L p Σ q ď } NS p B p z qq ´ B p z q} L p Σ q ` } B p z q ´ B p z q} L p Σ q ` } B p z q ´ NS p B p z qq} L p Σ q ď C ` } F B p z q } L p Σ q ` } F B p z q } L p Σ q ` a E p A ; M R q ˘ . It is standard to use the energy to bound the norm } F B p z q } L p Σ q . Hence one hassup ´ R ď s ď R sup ď t ,t ď R dist p u p s ` i t q , u p s ` i t qq ď C a E p A ; M R q . This proves (4.20). Then together with (4.19) the lemma is proved. (cid:3)
Consequences of the annulus lemma and the diameter estimate.
We firstprove the asymptotic behavior of solutions to the ASD equation.
Theorem 4.11 (Asymptotic Behavior) . Let A be a solution to the ASD equation over M “ R ˆ M . Then there exists a point p a ´ , a ` q P p ι ˆ ι q ´ p ∆ R Σ q – L M double whoseimage in R Σ is denoted by b such that the following conditions hold.(a) In the quotient topology of the configuration spaces A p P q{ G p P q one has lim s Ñ`8 r A | t˘ s uˆ M s “ a ˘ P L M Ă A p P q{ G p P q . (b) In the quotient topology of the configuration space A p Q q{ G p Q q one has lim z Ñ8 r A | t z uˆ Σ s “ b P R Σ Ă A p Q q{ G p Q q . Proof.
Denote the restriction of A to Ann ` p r,
8q ˆ
Σ by d H ` φds ` ψdt ` B . We firstprove the convergence of the gauge equivalence class r B p z qs . By the strong Uhlenbeckcompactness for Yang–Mills connections, we know that for each sequence z i Ñ 8 , thereis a subsequence (still indexed by i ) for which r B z i s converges in A p Q q{ G p Q q to a limit in R Σ . We would like to show that the subsequential limit is unique. By the annulus lemma(Proposition 4.5), there exist C ą δ P ą r ě E p A ; M (cid:114) M r q ď Cr ´ δ P . (4.21)Moreover, we claim that lim r Ñ8 } F A } L p M (cid:114) M r q “ . Indeed, if this is not the case, then a nontrivial ASD instanton over C ˆ Σ, a nontrivial ASDinstanton over R ˆ M , or a nontrivial R -instanton bubbles off at infinity, contradicting(4.21). Then by the diameter estimates (Lemma 4.9 and Lemma 4.10), there exist aconstant C ą r such that for all τ ě log r , there holdsdiam ` A ; M e τ ´ ,e τ ` ˘ ď C b E p A ; M e τ ´ ,e τ ` q ď Ce ´ δP τ (this is because the half annulus Ann ` p e τ ´ , e τ ` q can be covered by a fixed numberof half disks and a fixed number of disks with radii being comparable to e τ which arecontained in the half annulus Ann ` p e τ ´ , e τ ` q ). Hencelim r Ñ8 diam p A ; M (cid:114) M r q “ . Therefore the subsequential limit of r B p z qs in R Σ is unique. Denote the limit by b .Now we prove the convergence of r A | t s uˆ M s as s Ñ ˘8 . Abbreviate the restriction A t s uˆ M by A s . First, the finiteness of energy implies thatlim s Ñ˘8 } F A } L pr s ´ ,s ` sˆ M q “ . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 34
Hence by Uhlenbeck compactness, for any sequence s i Ñ ˘8 there is a subsequencefor which the sequence r A s i s converges to a limit in L M Ă A p P q{ G p P q . Then anysubsequential limit must be in ι ´ p b q where ι : L M í R Σ is the Lagrangian immersion.We would like to show that as s Ñ `8 or s Ñ ´8 the subsequential limit of r A s s isunique. Indeed, if there are two sequences s i , s i Ñ `8 such that r A s i s and r A s i s convergesto two different preimages of ι ´ p b q , denoted by a ` , a , then since the configurationspace A p P q{ G p P q is Hausdorff (see Lemma 4.12 below), one can choose two disjointneighborhoods U, U of a ` and a and a sequence s i Ñ `8 with r A s i s R U Y U . Thena subsequence of r A s i s converges to a limiting flat connection different from a ` and a .However since b has at most two preimages, this cannot happen. Hence the subsequentiallimit of r A s s is unique and hence r A s s converges to a limit a ˘ as s Ñ ˘8 . (cid:3) The following lemma is used in the previous proof.
Lemma 4.12.
Let M be a three-manifold with boundary and P Ñ M be an SO p q -bundle. Then the configuration space A p P q{ G p P q is Hausdorff with respect to the quotienttopology induced from the C -topology of A p P q .Proof. For A , A P A p P q , definedist C p A , A q : “ inf g P G p P q } g ˚ A ´ A } C p M q . (4.22)This clearly descends to a symmetric function on the quotient A p P q{ G p P q satisfyingthe triangle inequality. We claim that this is a metric, namely, if A and A are notgauge equivalent, then there is some δ ą C p A , A q ą δ . Suppose this isnot the case, then there exists a sequence of smooth gauge transformations g i such that } g ˚ i A ´ A } C p M q Ñ
0. Since g i takes value in a compact group SU p q , this implies thata subsequence of g i converges to a continuous gauge transformation g on P . Moreover,in the weak sense, g ˚ A “ A . Since both A and A are smooth, g is also smooth andhence A is gauge equivalent to A , which contradicts our assumption. Hence dist C is ametric on the configuration space and the lemma is proved. (cid:3) Theorem 4.11 is the analogue of the following results about the asymptotic behavior ofinstantons over C ˆ Σ. Theorem 4.13. [DS94, Proof of Theorem 9.1]
Let A be an ASD instanton over p C (cid:114) B R q ˆ Σ for some R ą which can be written as d C ` φds ` ψdt ` B p z q . Then thereis a gauge equivalence class of flat connections b P R Σ Ă A p Q q{ G p Q q such that in thequotient topology one has lim | z |Ñ8 r B p z qs “ b . Definition 4.14.
In the situation of Theorem 4.11 resp. Theorem 4.13, the point p a ´ , a ` q P L N resp. b P R Σ is called the evaluation at infinity of the solution A tothe ASD instanton over M resp. p C (cid:114) B R q ˆ Σ.Finally we have the energy quantization property for instantons over M “ R ˆ M . Theorem 4.15.
There exists (cid:126) ą which only depends on the bundle P Ñ M and whichsatisfies the following property. For any ASD instanton A over M there holds E p A q P t u Y r (cid:126) , `8q . Proof.
We claim that (cid:126) being the (cid:15) of Proposition 4.7 satisfies the condition of thistheorem. Indeed, suppose there is an ASD instanton over M with E p A q ă (cid:126) . For all OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 35 r ą R “ e s r with s sufficiently large, the restriction of A to M R satisfies thehypothesis of Proposition 4.7. Then E p A ; M r q “ E p A ; M Re ´ s q ď Ce ´ δ P s E p A ; M R q ď C(cid:15)e ´ δ P s . As s can be arbitrarily large, it follows that E p A ; M r q “
0. Therefore A is a trivialsolution. (cid:3) Compactness modulo energy blowup
In this section we prove a compactness theorem modulo bubbling. It is the analogueof Theorem 3.3 for the domain being M “ R ˆ M . We first set up the problem. Recallthat for any open subset S Ă H , M S denotes M S “ p S ˆ Σ q Y pB S ˆ M q where the two parts are glued over the common boundary B S ˆ Σ. Let S i Ă H be anexhaustive sequence of open subsets. They may or may not intersect the boundary of H . Let ρ i Ñ `8 be a sequence of positive numbers diverging to infinity. Recall that ϕ ρ : H Ñ H is the map corresponding to multiplying by ρ . In the proof of the maintheorem of this paper we will only use the case that S i “ H . Theorem 5.1.
Suppose A i is a sequence of ASD instantons on M ϕ ρi p S i q with lim sup i Ñ8 E p A i ; M ϕ ρi p S i q q ă `8 . Then there exist a subsequence (still indexed by i ) and a holomorphic map ˜ u “ p u , W , γ , m q , from H to R Σ with mass (see Definition 2.7) satisfying the following conditions.(a) For each z P H , one has the convergence lim r Ñ lim i Ñ8 E ` A i ; M B ` ρir p z q ˘ “ m p z q . (5.1) (b) Over ϕ ρ i p S i q ˆ Σ we write A i “ d H ` φ i ds ` ψ i dt ` B i . Then for any precompactopen subset K Ă H (cid:114) W , for i sufficiently large, B i p z q is in the domain of theNarasimhan–Seshadri map NS (see Definition 2.16) for all z P ϕ ρ i p K q , henceprojects down to a sequence of holomorphic maps u i : ϕ ρ i p K q Ñ R Σ . Further u i ˝ ϕ ρ i converges to u | K in the sense of Definition 2.5.(c) There is no energy lost in the following sense: lim R Ñ8 lim i Ñ8 E ` A i ; M ρ i R ˘ “ E p u q ` ż H m . (5.2)These two theorems motivates the following notion of convergence. Definition 5.2.
Let A i be a sequence of ASD instantons over M . Let ˜ u be a holo-morphic map from S to R Σ with mass in the sense of Definition 2.7. Suppose t ρ i u be asequence of positive numbers diverging to . A i is said to converge to u along with t ρ i u if conditions (a), (b), and (c) of Theorem 5.1 hold. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 36
Energy blowup threshold.
First we define the notion of energy blowup.
Definition 5.3 (Energy blowup) . Let ρ i , A i satisfy the assumptions of Theorem 5.1. Foreach w P H , we say that energy blows up at w iflim r Ñ lim sup i Ñ8 E ` A i ; M ϕ ρi p B ` r p w qq ˘ ą . Lemma 5.4 (Energy blowup threshold) . There exists (cid:126) ą satisfying the followingproperty. Let ρ i and A i be as in Theorem 5.1. Then for all w P H there holds lim r Ñ lim sup i Ñ8 E ` A i ; M ϕ ρi p B ` r p w qq ˘ P t u Y r (cid:126) , `8q . (5.3) Proof.
When energy concentrates at w P Int H , (5.3) follows from the bubbling analysisin [DS94, Proof of Theorem 9.1] (or via Proposition 4.8 as argued below). We consider thecase when w P B H . We claim that (cid:126) being the (cid:15) of Proposition 4.7 satisfies the property.Indeed, suppose on the contrarylim r Ñ8 lim sup i Ñ8 E p A i ; M ϕ ρi p B ` r p w qq q “ a ă (cid:15). Then there exists r ą i Ñ8 E p A i ; M ϕ ρi p B ` r p w qq q ă (cid:15). Then by Proposition 4.7 for r ! r and sufficiently large i , one has E p A i ; M ϕ ρi p B ` r p w qq q ď C ˆ rr ˙ δ P . This contradicts the assumption that energy concentrates at w . (cid:3) One can use the threshold to select a subsequence for which energy concentrationhappens at only finitely many points. Hence one can assume for a subsequence (stillindexed by i ) energy concentrates at points of a finite subset W Ă H and a bubblingmeasure m : H Ñ r , `8q supported over W satisfying (5.1).5.2. Constructing the limiting holomorphic curve.
Next we construct the limitingholomorphic curve. For any precompact open subset K Ă H (cid:114) W (which may intersect the boundary of H ), we claim thatlim i Ñ8 sup z P ϕ ρi p K q } F B i p z q } L p Σ q “ . Indeed, if this is not true, then some subsequence will bubble off an R -instanton, aninstanton over C ˆ Σ, or an instanton over M , contradicting that no energy concentratesat points away from W . Then for i sufficiently large, B i p z q is in the domain of theNarasimhan–Seshadri map NS : A , (cid:15) p Q q Ñ R Σ . By Proposition 2.17, the map u i : ϕ ρ i p K q Ñ R Σ , u i p z q : “ NS p B i p z qq is holomorphic. Denote the reparametrized holomorphic maps by u i : “ u i ˝ ϕ ρ i : K Ñ R Σ . One can choose a subsequence (still indexed by i ) and an exhausting sequence of precom-pact open subsets K i Ă H (cid:114) W such that u i is defined over K i . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 37
Lemma 5.5.
There is a subsequence of u i which converges to a holomorphic map u : H (cid:114) W Ñ R Σ in the sense of Definition 2.5. Moreover, the energy density function e ρ i of A i converges to the energy density of u in C p Int H (cid:114) W q .Proof. By Theorem 3.3, a subsequence of u i : K i Ñ R Σ (still indexed by i ) converges toa holomorphic map u : Int H (cid:114) W Ñ R Σ in C loc p Int H (cid:114) W q and the rescaled energydensity function converges in C p Int H (cid:114) W q to the energy density of u . To provethe boundary convergence, we need to verify that there is no diameter blowup along theboundary. Indeed, by the fact that no energy blowup happens in H (cid:114) W and Lemma4.10, there holds lim r Ñ lim sup i Ñ8 diam p u i p B ` r p z qqq “ , @ z P B H (cid:114) W . Then by Proposition 2.6 the map u extends continuously to the boundary and u i converges to u in the sense of Definition 2.5. (cid:3) We would like to use Gromov’s removal of singularity theorem to extend the limitingholomorphic map over the bubbling points. We first show that the limit u satisfies theLagrangian boundary condition. Lemma 5.6.
The holomorphic map u maps the boundary of H (cid:114) W into ι p L M q .Proof. For each s P B H (cid:114) W , denote A i ; s “ A | t s uˆ M . Since there is no energy blow upat s , one has lim i Ñ8 } F A i ; s } L p M q “ . Then there is a subsequence of A i ; s (still indexed by i ) which converges modulo gaugetransformation to a flat connection A ; s on M . Since the map induced from boundaryrestriction A p P q{ G p P q Ñ A p Q q{ G p Q q is continuous, one haslim i Ñ8 r B i ; s s “ lim i Ñ8 r A i ; s | Σ s “ ´ lim i Ñ8 r A i ; s s ¯ | Σ P ι p L M q . On the other hand, since } F B i ; s } L p Σ q Ñ
0, the point u i p s q P R Σ has a flat connectionrepresentative which is arbitrarily L -close to B i ; s . Since u i p s q converges to u p s q , u p s q agrees with the above limit which is in ι p L M q . (cid:3) Lemma 5.7.
There is a subsequence (still indexed by i ) such that for all s P B H (cid:114) W ,the sequence r A i ; s s converges in A p P q{ G p P q to a flat connection on P Ñ M .Proof. For each connected component I α Ă B H (cid:114) W , pick one point s α P I α . Then forthe finitely many chosen points z α , we can find a subsequence of A i , still indexed by i ,such that r A i ; s α s converges to r A s α s for all α . Then consider the sets I ˚ α “ ! s P I α | r A i ; s s converges in A p P q{ G p P q ) . Then s α P I ˚ α . Notice that, by the proof of Lemma 5.6, for each s P I ˚ α one has u p s q “ ι p lim i Ñ8 r A i,s sq . (5.4)Moreover, if u p s q is not a double point of ι p L M q , then s P I ˚ α . The lemma follows if wecan show that I ˚ α is both open and closed.We prove I ˚ α is open. Suppose this is not the case. Then there exist s ˚ P I ˚ α and asequence s k R I ˚ α with s k Ñ s ˚ . Since u is continuous, u p s k q converges to u p s ˚ q . Since OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 38 u p s k q is a double point for all k and the set of double points is discrete, u p s ˚ q “ u p s k q for sufficiently large k . Assume ι ´ p u p s ˚ qq “ t a , a u Ă L M and assume that r A i ; s ˚ s converges to a . Then for each k , there is a subsequence of r A i ; s k s which converges to a . By Lemma 4.12 which says the space A p P q{ G p P q is Hausdorff, we can choosedisjoint open neighborhoods U , U Ă A p P q{ G p P q . Then for each k , we can choose i k inductively such that 1) i k ` ą i k ; 2) r A i k ; s ˚ s P U ; 3) r A i k ; s k s P U . Then there existsa point w k between s ˚ and s k such that r A i k ; w k s R U Y U . Hence a subsequence (stillindexed by k ) of r A i k ; w k s converges to a point in L M which is neither a or a . Denote r B i k ; w k s “ r A i k ; w k s| B M . Then this convergence implieslim k Ñ8 u i k p w k q “ lim k Ñ8 r B i k ; w k s R ι p U Y U X L M q (5.5)which is different from u p s ˚ q . This contradicts the convergence u i towards u and thecontinuity of u . This proves the openness of I ˚ α .We prove I ˚ α is closed. Suppose this is not the case. Then there exists s ˚ R I ˚ α and a sequence s k P I ˚ α with s k Ñ s ˚ . Then u p s ˚ q is a double point of ι p L M q . Let U , U Ă A p P q{ G p P q be disjoint open neighborhoods of the two preimages of u p s ˚ q .Then there must be a subsequence of s k (still indexed by k ), and one of U and U , say U , such that for all k , lim i Ñ8 r A i ; s k s P U . By (5.4) and the non-convergence of r A i ; s ˚ s , there must also be a subsequence of i (stillindexed by i ), such that lim i Ñ8 r A i ; s ˚ s P U . Then for each k , one can find i k such that r A i k ; s k s P U , r A i k ; s ˚ s P U . Choose a sequence i k such that i k ` ą i k . Then since U and U are disjoint, one canfind for each k a point w k between z ˚ and z k with r A i k ; w k s R U Y U . Denote r B i k ; w k s “ r A i k ; w k s| B M . Then (5.5) still holds and one derives the same contra-diction. This proves the closedness of I ˚ α . (cid:3) It follows from Lemma 5.7 that the boundary map can be defined.
Corollary 5.8.
There exist a map γ : B H (cid:114) W Ñ L M and a subsequence (still indexedby i ) such that lim i Ñ8 r A i ; s s “ γ p s q , @ s P B H (cid:114) W (5.6) and u | B H (cid:114) W “ ι ˝ γ . Lemma 5.9. γ is continuous.Proof. One only needs to show the continuity at points whose image is a double point of ι p L M q . The argument is similar as in the proof of Lemma 5.7, using the Hausdorffness ofthe configuration space A p P q{ G p P q . The details are left to the reader. (cid:3) Corollary 5.10.
The holomorphic map u : H (cid:114) W Ñ R Σ extends to a holomorphicmap with immersed Lagrangian boundary condition with mass, denoted by ˜ u “ p u , γ , W , m q . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 39
Proof.
For each point w P W X Int H , by Gromov theorem of removal of singularity, u extends smoothly over w . On the other hand, given w P W X B H and considera punctured neighborhood of w that is biholomorphic to a strip p , `8q ˆ r , π s withcoordinates p s, t q . Then by the standard elliptic estimate for holomorphic maps withLagrangian boundary condition, one can prove that the length of the paths u p s, ¨q converges to zero. Hence the pair of points p γ p s, q , γ p s, π qq is either close to thediagonal of L M or is close to an ordered double point. In either case one can define a localaction for the path u p s, ¨q and this path satisfies the isoperimetric inequality similar to[MS04, Theorem 4.4.1 (ii)]. The isoperimetric inequality implies that as s Ñ `8 , thepaths u p s, ¨q converge to a constant path. Hence u extends continuously to w . (cid:3) Conservation of energy.
The last step of proving Theorem 5.1 is to verify theconservation of energy.
Lemma 5.11.
There holds lim R Ñ8 lim i Ñ8 E p A i ; M ρ i R q “ E p u q ` ÿ w P W m p w q . Proof.
To simplify the notations, assume that W contains only one point t u P B H .The proof of the general case can be derived easily from the proof of the special case,which we are going to present.Recall the closed three-manifold N and the SO p q -bundle P N Ñ N “ M double . Forany positive number r , the restriction of A i to B M ρ i r can be identified with a piecewisesmooth connection A i ; r on P N Ñ N . Then by the energy identity (Lemma 2.1) and thedefinition of energy concentration, one haslim R Ñ8 lim i Ñ8 E p A i ; M ρ i R q “ m p q ` lim r Ñ ,R Ñ8 lim i Ñ8 CS pr A i ; r s , r A i ; R sq where the last term is the Chern–Simons functional associated to the four-manifold withboundary M r,R (and the SO p q -bundle over it). On the other hand, for each pair p r, R q ,the holomorphic map u defines piecewise smooth connections A r , A R P A p . s . p P N q whosegauge equivalence classes are well-defined. Then by Proposition 2.18, one has E p u q “ lim r Ñ ,R Ñ8 CS pr A r s , r A R sq . Then by the property of Chern–Simons functional, to prove (5.2), it suffices to show thatlim r Ñ lim i Ñ8 CS pr A r s , r A i ; r sq “ R Ñ8 lim i Ñ8 CS pr A R s , r A i ; R sq “ . (5.8)We will only prove (5.7) as follows. The proof of (5.8) is similar.First, the evaluation of u at the origin defines a gauge equivalence class of flatconnections r A s P A p . s . p P N q{ G p . s . p P N q . From the finiteness of the energy of u one cansee that lim r Ñ CS pr A s , r A r sq “ . Hence it suffices to prove thatlim r Ñ lim i Ñ8 CS pr A s , r A i ; r sq “ . (5.9)Second, we would like to identify the relative Chern–Simons action CS pr A s , r A i ; r sq with the local action defined by Definition 4.2. OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 40
Claim.
For any ε ą
0, there exists r ε ą r P p , r ε q , one hassup r ď r ď r ε lim sup i Ñ8 ´ } F A i ; r } L p N ´ Y N ` q ` l p A i ; r q ¯ ď ε. Proof of the claim.
The smallness of } F A i ; r } L p N ´ Y N ` q follows from the fact that no energyblows up away from the origin and the (cid:15) -regularity of the Yang–Mills equation. Thesmallness of l p A i ; r q follows from the annulus lemma (Proposition 4.5) and the diameterestimate of Lemma 4.10. End of the proof of the claim.
Then for each r P p , r ε s , for sufficiently large i , the p “ A ˚ i ; r on P N satisfying } A i ; r ´ A ˚ i ; r } W , p N ´ Y N ` q ď Cε.
Then by the Sobolev embedding W , ã Ñ C in dimension three, for any given δ ą
0, for ε sufficiently small, r sufficiently small, and i sufficiently big, one hasdist C pr A i ; r | N ˘ s , r A ˚ i ; r | N ˘ sq ă δ (here dist C is defined by (4.22)). On the other hand, by the property of the pseudoholo-morphic curve u and its boundary map γ , for r sufficiently small, one hasdist C p γ p˘ r q , r A | N ˘ sq ă δ. Further, by the convergence r A i ; ˘ r | N ´ s Ñ γ p˘ r q , for sufficiently large i , one hasdist C pr A i ; r | N ˘ s , γ p˘ r qq ă δ. It follows that dist C pr A ˚ i ; r | N ˘ s , r A | N ˘ sq ă δ. Therefore, if δ is small enough, r A ˚ i ; r s and r A s are in the same connected component of L N . Hence CS pr A s , r A i ; r sq “ CS pr A ˚ i ; r s , r A i ; r sq “ F loc p A i ; r q . Then by the first version of the isoperimetric inequality (Theorem 4.3) one haslim r Ñ lim sup i Ñ8 | F loc p A i ; r q| ď C lim sup r Ñ lim sup i Ñ8 ´ } F A i ; r } L p N q ` l p A q ¯ “ . Hence (5.9) and therefore (5.7) follow. (cid:3) Stable scaled instantons
In this section we define the Gromov–Uhlenbeck convergence for ASD instantons. Infact we define a notion called stable scaled instantons , which is a combination of stablemaps in symplectic geometry and “instantons with Dirac measures” in gauge theory. Thecombinatorial model of trees is also used in the study of the vortex equation (see forexample [WX17] [WX]) and holomorphic quilts (see [BW18]).In this and the next sections, we will emphasize on the discussion of instantons over R ˆ M . The situation for instantons over C ˆ Σ can be dealt with similarly and most ofthe details will be left to the reader.
OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 41
Trees.
Let us fix a few notations. A tree, usually denoted by Γ, consists of a setof vertices V Γ and a set of edges E Γ . One can associate to each tree a 1-complex whose0-cells are vertices and whose 1-cells are edges. A rooted tree is a tree Γ together with adistinguished vertex v P V Γ called the root . A rooted subtree of a rooted tree p Γ , v q is a subtree which contains v . A ribbon tree is a tree Γ together with an isotopy classof embeddings of Γ into the complex plane. A based tree is a tree Γ with a rootedsubtree Γ with Γ equipped with the structure of a ribbon tree. The subtree Γ is called the base of Γ. A based tree can be used to model stable holomorphic disks such that verticesin the base correspond to disk components and vertices not in the base correspond tosphere components.We only consider rooted trees and often skip the term “rooted.” Notice that the root v P V Γ induces a natural partial order ď among all vertices: v ď v if v is closer to theroot v . We write v ą v if v ď v and v , v are adjacent; in this case we denote thecorresponding edge by e “ e v ą v P E Γ . Definition 6.1.
Consider the set t , ordered as 1 ď 8 . A scaling on a based tree Γis a map s : V Γ Ñ t , satisfying the following condition. ‚ Denote V “ s ´ p q and V Γ “ s ´ p8q . Then V Γ forms a (possibly empty) rootedsubtree of Γ and vertices in V are all disconnected from each other.A based tree Γ with a scale s is called a scaled tree .Each vertex of a scaled tree is supposed to support an instanton or a holomorphic mapover a certain domain, which we specify as follows. Consider a scaled tree Γ “ p Γ , s q with possibly empty base Γ. For each v P V Γ , define M v “ R ˆ M , S v “ H , and S v “ H Y t8u – D ; for each v P V Γ (cid:114) V Γ , define M v “ C ˆ Σ, S v “ C , and S v “ C Y t8u – S . For each v , there is an SO p q -bundle P v Ñ M v specified previously. Definition 6.2. A stable scaled instanton modelled on a scaled tree p Γ , s q consists ofa collection C “ ´(cid:32) p a v , m v q | v P V ( , (cid:32) w e | e P E Γ ( , (cid:32) u v “ p u v , W v , γ v q | v P V Γ (¯ where the symbols denote the following objects. ‚ For each v P V , a v is a gauge equivalence class of ASD instantons on P v Ñ M v and m v is a positive measure on M v with finite support. ‚ For each edge e “ e v ą v P E Γ , w e is a point of S v . ‚ For each v P V Γ , W v is the subset of S v defined by W v : “ t w e | e “ e v ą v P E Γ u (6.1)and u v is a holomorphic map from S v to R Σ with boundary in ι p L M q (see Definition2.7).These objects also satisfy the following conditions.(a) If e “ e v ą v P E Γ , namely both v and v are in the base, then w e P B S v .(b) For every v P V Γ , the collection of points defining W v in (6.1) are distinct.(c) The measure m v takes value in 4 π Z .(d) (Matching Condition) For each interior edge e “ e v ą v P E Γ (cid:114) E Γ the evaluationat infinity of a v or u v , which is a point of R Σ , is equal to the evaluation of u v at w e . For each boundary edge e “ e v ą v P E Γ the evaluation at infinity of a v or u v , which is a point in L N – ∆ L M Y R L M is equal to the transpose of ev w e p u v q (see Definition 2.7 and Definition 4.14 for the definitions of different evaluations.) OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 42 (e) (Stability Condition) If a v has zero energy, then m v ‰
0; if v P V Γ and u is aconstant map, then the number of boundary nodal points plus twice of the interiornodal points attached to S v is at least two; if v P V Γ (cid:114) V Γ and u v is a constantmap, then the number of nodal points on S v is at least two.A typical configuration of a stable scaled instanton has been shown in Figure 1.4.One can see that on the combinatorial level a stable scaled instanton is very similarto a stable affine vortex over C (when Γ has an empty base) or over H (when Γ has anonempty base). These two constructions appeared in [Zil14] and [WX17] respectively.We can define a notion of equivalence among stable scaled instantons by incorporatingthe translation symmetry of ASD instantons and the conformal invariance of holomorphicmaps. The details are left to the reader. It is then easy to check that the automorphismgroup of a stable scaled instanton is finite.6.2. Sequential convergence.
Now we define the notion of convergence of a sequenceof ASD instantons over M “ R ˆ M towards a stable scaled instanton. Let C “ ´(cid:32) p a v , m v q | v P V ( , (cid:32) w e | e P E Γ ( , (cid:32) u v | v P V Γ (¯ be a stable scaled instanton modelled on a based scaled tree p Γ , s q . For each v P V Γ , wehave the set of nodes W v Ă S v defined by (6.1). Then we can define a measure m v on S v supported in W v as follows. For each w e P W β with e “ e v ą v , the mass m v at w e is thesum of the energy of all components in C labelled by v P V Γ with v ě v . Denote˜ u v “ p u v , m v q which is a holomorphic curve with mass (see discussion after Definition 2.7). Definition 6.3 (Convergence towards stable scaled instantons) . Let a i “ r A i s be asequence of gauge equivalence classes of ASD instantons on P Ñ M . Let p Γ , s q be abased scaled tree. Let C “ ´(cid:32) p a v , m v q | v P V ( , (cid:32) w e | e P E Γ ( , (cid:32) u v | v P V Γ (¯ be a stable scaled instanton modelled on p Γ , s q . We say that a i converges (modulo gaugetransformation) to C if the following conditions are satisfied.(a) For each v P V , there exist a sequence of real translations ϕ i,v p z q “ z ` Z i,v such that ϕ ˚ i,v a i converges in the Uhlenbeck sense to p a v , m v q over the manifold M v “ R ˆ M (see Definition 2.2).(b) For each v P V (cid:114) V , there exist a sequence of complex translations ϕ i,v p z q “ z ` Z i,v satisfying the following properties.(i) Im Z i,v Ñ `8 ;(ii) Viewing ϕ i,v as a diffeomorphism from C ˆ Σ to itself, for all R ą ϕ ˚ i,v p a i | ϕ i,v p B R qˆ Σ q converges to p a v | B R ˆ Σ , m v | B R ˆ Σ q in the Uhlenbeck sense.(c) For each v P V Γ , there exists a sequence of real affine transformations ϕ i,v p z q “ ρ i,v p z ` Z i,v q satisfying the following properties.(i) ρ i,v P R , Z i,v P R , ρ i,v Ñ `8 ;(ii) Denoting the translation z ÞÑ z ` Z i,v by ψ i,v and viewing it as a diffeomor-phism from M to itself, ψ ˚ i,v a i converges to ˜ u v along with t ρ i,v u in the senseof Definition 5.2.(d) For each v P V Γ (cid:114) V Γ , there exist a sequence of complex affine transformations ϕ i,v p z q “ ρ i,v p z ` Z i,v q satisfying the following properties.(i) ρ i,v P R , Z i,v P C , ρ i,v Ñ `8 , and Im Z i,v Ñ `8 ; OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 43 (ii) Denoting the translation z ÞÑ z ` Z i,v to be ψ i,v and viewing it as a diffeomor-phism from C ˆ Σ to itself, then for all sufficiently big R ą
0, the sequence ofpullbacks ψ ˚ i,v p a i | ϕ i,v p B R qˆ Σ q , which are instantons over B ρ i,v R ˆ Σ, convergeto ˜ u v | B R along with t ρ i,v u in the sense of Definition 3.4.(e) The reparametrizations ϕ i,v mentioned above can all be viewed as M¨obius trans-formations on the complex plane. One can also view all S v (being either H or C ) as subsets of C . For each edge e “ e v ą v P E Γ , the map ϕ ´ i,v ˝ ϕ i,v convergesuniformly with all derivatives on compact sets to the constant w e P S v Ă C .(f) There is no energy lost, i.e.,lim i Ñ8 E p A i q “ ÿ v P V Γ E p u v q ` ÿ v P V ´ E p a v q ` | m v | ¯ . Now we state the precise form of the main theorem (Theorem 1.4) of this paper.
Theorem 6.4.
Given a sequence of ASD instantons on R ˆ M with uniformly boundedenergy, there exists a subsequence which converges to a stable scaled instanton in the senseof Definition 6.3. The rest of this paper provides a proof of Theorem 6.4.
Remark . It is easy to extend our definitions and our compactness theorem to thecase of instantons over C ˆ Σ. For example, when Γ is a scaled tree with empty base,the object defined in Definition 6.2 is a possible limiting configuration of a sequence ofinstantons over C ˆ Σ. One can then prove the corresponding compactness theorem(Theorem 1.5) for such instantons in a routine way.7.
Proof of the compactness theorem
Now we start to prove the compactness theorem (Theorem 6.4). The basic strategy issimilar to the proof of Gromov compactness for pseudoholomorphic curves used in [MS04]:starting from the root component, we construct each component of the limiting objectinductively. The induction is based on the “soft rescaling” argument in which one canverify the stability condition of the limiting object. Then one uses the annulus lemma toshow that the limit object satisfies the matching condition.Before we start, we remark on certain conventions and simplifications which we willfollow in our proof.
Remark . (a) Reset the value of (cid:126) ą (cid:15) of the annulus lemma(Proposition 4.5 and Corollary 4.6), the threshold of energy concentration given byLemma 5.4, the minimal energy of nonconstant holomorphic spheres in R Σ , theminimal energy of nonconstant holomorphic disks in R Σ with boundary in ι p L M q having at most two switching points, the minimal energy of nonconstant ASDinstanton over C ˆ Σ (the existence of the minimal energy is proved in [DS94,Proof of Theorem 9.1] and [Weh06]), and the minimal energy of a nontrivial ASDinstanton over M given by Theorem 4.15.(b) From now on we fix a sequence of ASD instantons A i P A p P q satisfyingsup i E p A i q ă `8 . By Theorem 4.15 and the above item of this remark, we may assume that E p A i q ě (cid:126) p@ i ě q . (7.1) OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 44 (c) It is also convenient to fix the ambiguity caused by the translation invariance. Foreach i ě Z P R , let R “ R p Z q be the minimal real number satisfying E ` A i ; M (cid:114) M B ` R p Z q ˘ “ (cid:126) . By the decay of energy, when Z approaches to ˘8 , the R p Z q satisfying theabove condition approaches to . It is also easy to see that R p Z q depends on Z continuously. Then one can choose for each i a number Z i (which might not beunique) such that R i “ R p Z i q is smallest possible. Then by using a translation of M in the R -direction, we may assume Z i “ i . Hence one has E ` A i ; M (cid:114) M R i ˘ “ (cid:126) . (7.2)Now we start our construction of the limiting object. By taking a subsequence, we mayassume that either R i stays bounded or diverges to infinity. We first prove that there isno energy concentrating in the area further away from M R i . Proposition 7.2.
For any ρ i with ρ i " R i , one has lim i Ñ8 E ` A i ; M (cid:114) M ρ i ˘ “ . (7.3) Proof.
Follows from the annulus lemma (Proposition 4.5). (cid:3)
Constructing the root component.
Now we are going to construct the rootcomponent of the limiting object. A simple situation is when R i is bounded. Then byTheorem 2.3, a subsequence of A i converges to an ASD instanton A with a bubblingmeasure m in the Uhlenbeck sense. Then pr A s , m q is the limiting object. Indeed,the scaled tree underlying this limiting object has only one vertex, and Proposition 7.2 isequivalent to the no-energy-lost condition of Definition 6.3.From now on we assume that R i is unbounded and we take a subsequence such that R i diverges to infinity. Then by applying Theorem 5.1 for R i and A i , one obtains asubsequence (still indexed by i ) and a holomorphic curve with mass ˜ u “ p u , m q from H to R Σ such that A i converges to ˜ u along with t R i u in the sense of Definition 5.2.Proposition 7.2 and (5.2) imply thatlim i Ñ8 E p A i q “ E p u q ` | m | . (7.4)We would like to show that this limiting component is stable. Lemma 7.3. If u is a constant map, the support of m contains either an interiorpoint of H , or contains at least two points in the boundary of H .Proof. The assumption (7.1) and the equality (7.4) imply that if u is constant, then W “ Supp m ‰ H . Suppose W Ă B H and contains only one element w . Then allenergy is concentrated at w , which means thatlim r Ñ lim i Ñ8 E ´ A i ; M (cid:114) M ϕ Ri p B ` r p w qq ¯ “ . Since the total energy is at least (cid:126) , there exists a sequence of r i Ñ i sufficiently large, E ´ A i ; M (cid:114) M ϕ Ri p B ` ri p w qq ¯ “ E ´ A i ; M (cid:114) M B ` riRi p R i w q ¯ “ (cid:126) . Since r i R i ! R i , this contradicts the choice of R i (see (c) of Remark 7.1). (cid:3) OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 45
Soft rescaling.
What we have done is constructing the root component of thelimiting stable scaled instanton. Next we inductively construct all other componentsusing the “soft-rescaling” method. The soft-rescaling method is similar to that of [MS04,Section 4.7], which was also used in [Fra08] for the compactness problem of holomorphicdisks, in [Zil14] for the compactness problem of affine vortices, and in [WX17] for theadiabatic limit of disk vortices. Let v denote the root component and m the bubblemeasure. Suppose W “ Supp m “ t w k | k “ , . . . , k u . For each w k P Supp m , one has thatlim r Ñ lim i Ñ8 E ´ A i ; ϕ R i p M B ` r p w k q q ¯ “ m p w k q ě (cid:126) . (7.5)Then choose r ą B r p w k q X W “ t w k u and for sufficiently big i there holds m p w k q ´ (cid:126) ď E ´ A i ; ϕ R i p M B ` r p w k q q ¯ “ E ´ A i ; M B ` Rir p R i w k q ¯ ď m p w k q ` (cid:126) . Then for each w P B r p w k q and i , one there exists r i p w q ą E ˆ A i ; ϕ R i p M B ` ri p w q p w q q ˙ “ m p w k q ´ (cid:126) . Notice that the minimal r i p w q depends continuously on w . Therefore, one chooses w i,k P B r p w k q such that r i,k : “ r i p w i,k q “ min w P B r p w k q r i p w q . (7.6)Since energy concentrates at w , w i,k converges to w k and r i p w i,k q converges to zero. Thepoint w i,k can be viewed as the point near w k where energy concentrates in the fastestrate (an analogue of the local maximum of the energy density).The following argument bifurcates in different cases. Suppose w i,k “ s i,k ` i t i,k . By choosing a suitable subsequence, we are in one of the following situations.
Case I.
Suppose we are in the situation where w k P Int H , lim i Ñ8 r i,k R i ă 8 . (7.7)Then choose a sequence ρ i Ñ 8 but grows slower than R i . Then when i is large, B ρ i p R i w i,k q Ă C is contained in the upper half plane. Define the sequence of translationson C by ϕ i,k p z q “ z ` R i w i,k . Via ϕ i,k , the sequence of restrictions of A i to B ρ i p R i w i,k q ˆ Σ can be identified with asequence A i,k of solutions to the ASD equation over B ρ i ˆ Σ. The total energy of A i,k isuniformly bounded, hence a subsequence (still indexed by i ) converges in the Uhlenbecksense to a limiting ASD instanton A k over M k : “ C ˆ Σ with a bubbling measure m k on M k .We prove that no energy is lost. Lemma 7.4.
There holds m p w k q “ E p A k q ` ż M k m k . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 46
Proof.
Choose (cid:15) ą
0. By assumption, there exists r (cid:15) ą i sufficiently large,there holds E ´ A i,k ; B R i r (cid:15) ˆ Σ ¯ ď m p w k q ` (cid:15). Then the Uhlenbeck convergence A i,k Ñ p A k , m k q implies that E p A k q ` | m k | “ lim R Ñ8 lim i Ñ8 E ´ A i,k ; B R ˆ Σ ¯ ď lim i Ñ8 E ´ A i,k ; B R i r (cid:15) ˆ Σ ¯ ď m p w k q ` (cid:15). On the other hand, we claim that when (cid:15) is small enough, there exists τ (cid:15) ą E ´ A i,k ; B τ (cid:15) ˆ Σ ¯ ě m p w k q ´ (cid:15). (7.8)If this is not true, then there exist a sequence τ i Ñ 8 such that E ´ A i,k ; B τ i ˆ Σ ¯ “ m p w k q ´ (cid:15). There are also a sequence τ i ą τ i which grows slower than R i such that E ´ A i,k ; B τ i ˆ Σ ¯ “ m p w k q ´ (cid:15). Hence E ´ A i,k ; p B τ i (cid:114) B τ i q ˆ Σ ¯ “ (cid:15). (7.9)On the other hand, choose R ą r i,k R i . Then one has E ´ A i,k ; p B R i r (cid:15) (cid:114) B R q ˆ Σ ¯ “ E ´ A i,k ; B R i r (cid:15) ˆ Σ ¯ ´ E ´ A i,k ; B R ˆ Σ ¯ ď E ´ A i,k ; B R i r (cid:15) ˆ Σ ¯ ´ E ´ A i,k ; B r i ,k R i ˆ Σ ¯ ď m p w k q ` (cid:15) ´ p m p w k q ´ (cid:126) q “ (cid:126) ` (cid:15). The conformal radius of B R i r (cid:15) (cid:114) B R diverges to infinity; one also has τ i { R Ñ 8 , τ i { R i r (cid:15) Ñ
0. Hence when (cid:15) is small enough, (7.9) contradicts Corollary 4.6. Thereforethe claim of the existence of τ (cid:15) satisfying (7.8) is true. Then one has E p A k q ` | m k | ě lim i Ñ8 E ´ A i,k ; B τ (cid:15) ˆ Σ ¯ ě m v p w k q ´ (cid:15). Since (cid:15) can be arbitrary small number, this lemma is proved. (cid:3)
The following lemma follows directly from Lemma 7.4.
Lemma 7.5. If A k is a trivial instanton, then m k ‰ . Case II.
Suppose we are in the situation where w k P B H , lim i Ñ8 p r i,k ` t i,k q R i ă 8 . Then choose a sequence ρ i Ñ 8 but grows slower than R i . Define the sequence of realtranslations by (recall s i,k “ Re w i,k ) ϕ i,k p z q “ z ` R i s i,k . Via ϕ i,k , the sequence of restrictions of A i to M B ` ρi p R i s i,k q can be identified with a sequence A i,k of solutions to the ASD equation over M B ` ρi . The energy of A i,k is uniformly bounded.Then a subsequence of A i,k converges in the Uhlenbeck sense to an ASD instanton A k over M k “ M “ R ˆ M with a bubbling measure m k . Similar to Lemma 7.4, one usethe annulus lemma (Proposition 4.5) for the case N “ M double to prove that no energy islost, i.e., m p w k q “ E p A k q ` ż M k m k . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 47
Similar to the above case, one has the following facts.
Lemma 7.6. If A k is trivial, then m k ‰ . Case III.
Suppose we are in the situation where w k P Int H , lim i Ñ8 τ i,k : “ lim i Ñ8 r i,k R i “ 8 . In this case, choose a sequence ρ i Ñ 8 which grows faster than τ i,k but slower than R i . Then when i is large, the disk B ρ i p R i w i,k q Ă C is contained in the upper half plane.Define a sequence of affine transformations ϕ i,k and a sequence of translations ψ i,k by ϕ i,k p z q “ τ i,k ˆ z ` w i,k r i,k ˙ , ψ i,k p z q “ z ` w i,k r i,k . Via ψ i,k , the sequence of restrictions of A i to B ρ i p R i w i,k q ˆ Σ can be identified witha sequence A i,k of solutions to the ASD equation over B ρ i ˆ Σ. By Theorem 3.3 asubsequence of A i converges modulo bubbling to a holomorphic map ˜ u k “ p u k , m k q withmass from C to R Σ along with the sequence t τ i,k u . Using similar method as provingLemma 7.4 one can prove that m p w k q “ E p u k q ` ż C m k . Moreover, we verify the stability condition as follows.
Lemma 7.7.
When u k is a constant, the support of m k contains at least two points.Proof. If this is not the case, then m k is supported at a single point z P C with m k p z q “ m p w k q . This implies thatlim r Ñ8 lim i Ñ8 E ´ A i,k ; B rτ i,k p z q ˆ Σ ¯ “ m k p z q “ m p w k q . Since m k p z q ě (cid:126) , there is a sequence δ i Ñ iE ´ A i,k ; B δ i τ i,k p z q ˆ Σ ¯ “ E ´ A i ; ϕ R i p M B r i p w i,k ` z q q ¯ “ m p w k q ´ (cid:126) . Here r i “ δ i τ i,k R i “ δ i r i,k which is smaller than r i,k . This contradicts the choice of w i,k and r i,k (see (7.6)). Hencethe support of m k contains at least two points. (cid:3) Case IV.
Suppose we are in the situation where w k P B H , lim i Ñ8 τ i,k : “ lim i Ñ8 R i p r i,k ` t i,k q “ 8 . (7.10)Define the sequence of affine transformations ϕ i,k and translations ψ i,k by ϕ i,k p z q “ τ i,k p z ` R i s i,k q , ψ i,k p z q “ z ` R i s i,k Denote the pull-back of A k via ψ i,k by A i,k , which are still ASD instantons over M .Without loss of generality, we may assume that R i s i,k “ i . Then by Theorem5.1, a subsequence of A i,k (still indexed by i q converges to a holomorphic map with mass˜ u k “ p u k , m k q from H to R Σ along with the sequence t τ i,k u . Moreover, one haslim R Ñ8 lim i Ñ8 E ´ A i,k ; M ϕ τi,k p B ` R q ¯ “ E p u k q ` ż H m k . OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 48
As before one can prove that the left hand side above is m p w k q , hence m p w k q “ E p u k q ` ż H m k . We verify the stability condition as follows.
Lemma 7.8.
Suppose u k is a constant map.(a) If lim i Ñ8 t i,k { r i,k ă 8 , then Supp m k contains at least two points;(b) If lim i Ñ8 t i,k { r i,k “ 8 , then Supp m k contains at least one point in the interior of H .Proof. Suppose lim i Ñ8 t i,k { r i,k ă 8 and Supp m k contains only one point z P H with m k p z q “ m p w k q . This implies thatlim r Ñ lim i Ñ8 E ´ A i,k ; M ϕ τi,k p B ` r p z qq ¯ “ m p w k q . Then there exists a sequence δ i Ñ E ´ A i ; M ϕ Ri p B ` τi,kδi { Ri p τ i,k z { R i qq ¯ “ E ´ A i,k ; M ϕ τi,k p B ` δi p z qq ¯ “ m p w k q ´ (cid:126) . However we have τ i,k δ i R i “ δ i p r i,k ` t i,k q ! r i,k , which contradicts the choice of r i,k (see (7.6)).On the other hand, suppose lim i Ñ8 t i,k { r i,k “ 8 . Then τ i,k " r i,k R i . Still assume forsimplicity that s i,k “ Re w i,k “
0. Then for any r ą
0, for i sufficiently large, the diskcentered at R i w i,k “ i R i t i,k of radius r i,k R i is contained in the disk centered at i τ i,k ofradius rτ i,k . This implies that m k is supported at the single point i P Int H . (cid:3) Bubble connects.
The above discussion of four different cases allows us to induc-tively construct the limiting object. The process stops after finitely many steps due tothe energy quantization phenomenon (see item (a) of Remark 7.1). This finishes theconstruction of a collection of ASD instantons with measures pr A v s , m v q , a collection ofholomorphic maps u v from S v to R Σ with boundary lying in ι p L M q . They satisfylim i Ñ8 E p A i q “ ÿ v P V ´ E p A v q ` | m v | ¯ ` ÿ v P V Γ E p u v q . Lemma 7.5, 7.6, 7.7, and 7.8 imply that the collection satisfies the stability condition ofDefinition 6.2. To show that the limiting object satisfies the definition of stable scaledinstantons (Definition 6.2), it remains to prove that this collection form a stable scaledinstanton, i.e., proving “bubbles connect” or more precisely, the matching condition ofDefinition 6.2. We only prove this condition for a boundary edge/node. The case forinterior edge/node can be proved in a similar way. Let e v ą v P E Γ be a boundary edge ofthe limiting graph Γ. Then by the construction of the limit, there are two sequences ofM¨obius transformations ϕ i,v p z q “ ρ i,v p z ` Z i,v q , ϕ i,v “ ρ i,v p z ` Z i,v q with Z i,v , Z i,v P R and ρ i,v " ρ i,v . By the translation invariance, one can assumethat Z i,v ”
0. Then one has the following lemma (recall the notion of the diameter inSubsection 4.3.)
OMPACTNESS FOR ASD EQUATION WITH TRANSLATION SYMMETRY 49
Lemma 7.9 (Bubble connects) . For sufficiently large s one has lim i Ñ8 sup z P Ann ` p e s ρ i,v ,e ´ s ρ i,v q } F A i } L pt z uˆ Σ q “ and lim s Ñ8 lim i Ñ8 diam p A i ; M e s ρ i,v ,e ´ s ρ i,v q “ . (7.12) Proof.
By the inductive construction of the limiting object, there exists s ą i , there holds E p A i ; M e s ρ i,v ,e ´ s ρ i,v q ď (cid:126) . Then the annulus lemma (Proposition 4.5), one haslim s Ñ8 lim i Ñ8 E p A i ; M e s ρ i,v ,e ´ s ρ i,v q “ . Then (7.11) follows from standard estimate of the ASD equation. Hence the diameterin the limit (7.12) is defined for sufficiently large s and sufficiently large i . Then (7.12)follows then from the diameter bound given by Lemma 4.9 and Lemma 4.10. (cid:3) The above lemma implies that the evaluations of the two sides of the node correspondingto the edge e v ą v (which are points in L N ) agree after being mapped to ι p L M q Ă R Σ . Onecan use the ASD equation on B Ann ` p e s ρ i,v , e ´ s ρ i,v q ˆ M and the fact that the energy ofthe solution restricted to this region shrinks to zero to prove that the evaluations are thesame point of L N (after transposing). This finishes the proof of the matching conditionfor the edge e v ą v .We declare that we have finished the proof of Theorem 6.4 (the same as Theorem 1.4).As we have addressed previously, the case of instantons over C ˆ Σ (Theorem 1.5) can beproved using a similar and more simplified method.
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Department of Mathematics, Texas A&M University, College Station, TX 77843 USA
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