aa r X i v : . [ m a t h . S G ] S e p GROMOV-UHLENBECK COMPACTNESS
MAX LIPYANSKIY Introduction
The Atiyah-Floer Conjecture.
In the early 1980’s Casson (see [16]) intro-duced a new invariant of 3-manifolds based on a ”count” of representations of thefundamental group of a 3-manifold to SU (2), or more generally a compact Lie group G . Let us briefly recall the basic idea. Let Y be a closed oriented 3-manifold and let Y = H + ∪ Σ H − be a Heegaard splitting of Y along a genus g surface Σ. Let M (Σ) be the charactervariety of of Σ. This is defined as the space of representations Hom ( π (Σ) , SU (2)) /SU (2)modulo conjugation by SU (2). As observed by Atiyah and Bott [2], M (Σ) is acompact K¨ahler manifold of dimension 6 g − π (Σ) → π ( H ± )induce injective maps L ± → M (Σ)where L ± is the set of representations of the free group π ( H ± ). In fact, each L ± hasdimension 3 g − L ± are Lagrangian for the natural symplectic formon M (Σ). By considering generic intersections of L ± (in fact, 1/2 of the number ofintersections), Casson was able to define a count of representations, in the case when Y is a integral homology sphere.Taubes [17] gave a gauge theoretic interpretation of Casson’s results. Let us brieflyrecall the basic idea. Let A be a connection on a trivial SU (2)-bundle over Y andlet F A be the curvature. Connections with vanishing curvature are called flat andare classified by their holonomy. Therefore, we have a natural correspondence be-tweeen flat connections (module gauge) and representations of π ( Y ) into SU (2). Byintroducing suitable perturbations Taubes defines a gauge-theoretic count of such flatconnections. In fact, the Casson invariant is an infinite dimensional analogue of Eulercharacteristic of the space of all connections modulo gauge. .2. Categorifications.
We have arrived at two different geometric descriptions ofthe Casson invariant - one based on gauge theory and the other on symplectic geom-etry. As the reader might anticipate, both descriptions of the invariant have categori-fications that express the invariant as the euler characteristic of a certain homologygroup. In both cases, the corresponding homology theory is due to the groundbreak-ing work of Floer.Let us first discuss the Lagrangian viewpoint. Given a pair of Lagrangians L ± , ina symplectic manifold M (Σ), Floer constructs a chain complex C ∗ ( L − , L + ) freelygenerated by the set L − ∩ L + . Given two intersection points x, y ∈ L − ∩ L + , thedifferential on C ∗ ( L − , L + ) counts holomorphic strips u : [0 , × R → M (Σ)with u (0 , · ) ∈ L − and u (1 , · ) ∈ L − . Let us denote the resulting groups by HF ∗ ( L − , L + )From the gauge theory perspective, we define the chain complex C ∗ ( Y ) as follows.The generators are given by flat connections A on Y . The differential, on the otherhand, is the signed count of solutions to the ASD equation ∗ F B = − F B on the 4-manifold Y × R . These are required to have finite energy and converge tospecified flat connections at the ends. Let us denote the resulting groups by I ∗ ( Y )Under the assumption of transversality, we note that the generators of the two chaincomplexes are identical. However, the corresponding homology theories are based onsolutions to nonlinear PDE’s in dim 2 and 4 and a priori do not appear to be related.We have the following: Atiyah-Floer Conjecture [1] : There exists an isomorphism I ∗ ( Y ) ∼ = HF ∗ ( L − , L + )The immediate problem with this conjecture is that the relevant group on the sym-plectic side has not been defined. This is related to the fact that the presence ofreducible representations cause singularities in the character variety (see however [5]).On the other hand, there are several ways of getting around this issue that leadto interesting and well defined groups. If b ( Y ) >
0, one approach is to considernontrivial U (2)-bundles over Y with odd c . This way, all flat connections are ir-reducible the relevant spaces have well defined groups. A proof of the analogue ofthe conjecture for the case of a mapping torus has been given in [7] using adiabaticlimits. This work is part of a series to prove the conjecture for a general Y with a ontrivial U (2)-bundle - thus covering all cases where the groups are well-defined.Our approach is not based on adiabatic techniques but rather develops an analyticsetting ((which we call Gromov-Uhlenbeck compactness) that combines the pseudo-holomorphic curves and ASD connections into a unified framework. As a consequenceof the general theory, one constructs an apriori mapΦ : I ∗ ( Y ) → HF ∗ ( L − , L + )that can be shown to be an isomorphism. The present work, is devoted to the analyticfoundations of such a theory. Applications to Floer homology will be addressedelsewhere. We hope, however, that the present techniques are of independent interestsand may be of use in other contexts where gauge theory and symplectic geometryinteract.1.3. Overview of the Results.
We give a brief outline of the main compactnessresults of this paper.Consider a compact Riemann surface Σ with complex structure j Σ and Kahler metric g Σ . Let E → Σ be a 2-dimensional complex vector bundle with odd c ( E ) ∈ H (Σ; Z ) ∼ = Z Let V → Σ be the bundle of traceless endomorphisms of E . Note that V has struc-ture group SO (3). We fix once and for all a unitary connection α det on det ( E ). Oncethis choice of α det is made, we have an identification between SO (3)-connections on V and U (2)-connections on E that induce α det on det ( E ).Let G (Σ) E be the group of U (2)-gauge transformations of E that descend to theidentity on det ( E ). G (Σ) has a natural induced action on the connections on V andwe let G (Σ) ⊂ G (Σ) V be its image. We may identify the action of G (Σ) with theaction of SO (3)-gauge transformations on V that lift to gauge transformations of E . Definition 1.
Let A be the affine space of SO (3) -connections on V . G (Σ) acts on A by g ∗ ( α ) = α + g − d α g Following Atiyah and Bott [2], we note that this action is Hamiltonian with momentmap µ : A → Ω (Σ; g )given by µ ( α ) = ∗ F α , where F α is the curvature of α . Let C ⊂ A be the set ofprojectively flat connections.
Definition 2.
Let M = µ − (0) / G (Σ) = A // G (Σ) n fact, M is a compact Kahler manifold (see [2]). M has a concrete descriptionin terms of representations of π (Σ). Pick a point p ∈ Σ. The space M is the spaceof representations π (Σ − p ) → SU (2) that have holonomy − I around p , moduloconjugation by SU (2). If we pick a standard homology basis { α i } gi =1 , { β i } gi =1 , wemay identify M with the space(1) { g i ∈ SU (2) , h i ∈ SU (2) | Π gi =1 [ g i , h i ] = − I } /SU (2)In general, given a symplectic manifold ( M, ω ) with a Hamiltonian group action G and corresponding moment map µ : M → Lie ( G )one may form the symplectic reduction at 0 by M//G = µ − (0) /G Provided G acts freely on µ − (0), ( M//G, ω
M//G ) inherits a symplectic structure from M . Let ( M//G ) − denote the symplectic manifold with the opposite form. There isa canonical Lagrangian L = µ − (0) ⊂ ( M//G ) − × M defined as the set of pairs ([ m ] , m ). Here, [ m ] = mG denotes the orbit of m . Thisgeneral construction applies to the case of interest where M = A and G = G (Σ). Definition 3.
Let
L ⊂ M × A be the set of pairs ([ α ] , α ) where α is a flat connectionon Σ . Matching Boundary Conditions.
Let B R ( p ) ⊂ C be the closed disk of radius R centered at the origin and let H + = { ( s, t ) ∈ C | s ≥ } be the positive half-plane. We define D + R as H + ∩ B R . Let D − R be the reflection ofthis disk in the t -axis. In general, ∂D + R = I R ∪ S R where I R = { (0 , t ) ∈ D + R } and S R = { ( s, t ) ∈ D + R | s + t = R } The interior of a disk D + R is the set of points with s + t < R and will be denotedby ˚ D + R .Consider a holomorphic map u : D − R → M and an ASD connection A on D + R × Σ. For each t ∈ I R , we may restrict A to the slice(0 , t ) × Σ. This gives us a map R A : I R → A (Σ) efinition 4. The pair ( u, A ) is said to be matched if at each t ∈ I R , u (0 , t ) = [ R A ( t )] where [ R A ( t )] denotes the gauge orbit of R A ( t ) . Note that this is precisely the condition that ( u (0 , t ) , R A ( t )) ∈ L for each t . If A was a holomorphic curve, this would amount to a Lagrangian boundary condition forthe pair (˜ u, A ) : D + R → A (Σ)where ˜ u ( s, t ) = u ( − s, t ). For convenience, we will refer to the pair ( u, A ) as definedon D + R × Σ. Let e ( s, t ) = 12 | du ( − s, t ) | + 12 Z Σ | F A | and E ( u, A ) = Z D + R e The following gives a key a priori estimate for matched pairs:
Theorem 1.
There exists ~ , C > with the following property. Let ( u, A ) be amatched pair on some D + R × Σ . If E ( u, A ) < ~ then e ( p ) ≤ C E ( u, A ) R − We now state our main compactness results. We will consider a sequence of matchedpairs ( u i , A i ) on D + R × Σ. By the regularity results of section 6, we may assume thatthe sequence consists of smooth elements.
Definition 5. A singular set S on D R × Σ is a finite collection of points x i ∈ ( D − R − ∂D − R ) , y i ∈ ( D R − ∂D R ) × Σ , z i × Σ ∈ ( I R − ∂I R ) × Σ . The z i × Σ are the boundary slices of S . Theorem 2.
Assume that we have a uniform bound E ( u i , A i ) < C . There exists asubsequence ( u j , A j ) and a singular set S with the following properties. Let K be acompact set in ˚ D − R − S and K be a compact set in ˚ D R × Σ − S . We have that u j converges in any C k norm on K and A j converges in any C k -norm on K . Finally,the energy loss at each singular point is at least ~ for some sufficiently small ~ > independent of the choice of sequence. Acknowledgement.
We wish to thank Tom Mrowka, Dusa McDuff and DennisSullivan for useful conversations. In addition, we would like to thank the SimonsCenter For Geometry and Physics for their hospitality while this work was beingcompleted. . Review of Sobolev Spaces
The Space L pk . In this work we will need to consider Sobolev spaces of func-tions with an infinite dimensional Banach space as target. Thus, we begin with abrief review of Sobolev spaces with the purpose of setting down notation as well asexplaining how the results extend with minimal effort to the infinite dimensional case.A general reference for Sobolev spaces is [9].Let M be a closed, oriented, Riemannian manifold of dimension d . For p > k anonnegative integer, one has the Sobolev space L pk ( M ) of real valued functions on M .These are defined by completing the space of smooth functions on M with respect tothe norm:(2) k X i =0 ||∇ i f || L p More generally, given a vector bundle V → M , one may consider the Sobolev space ofsections of V . The results of this section apply in this general context. We will oftenomit M from the notation when the domain is clear in a particular discussion. Let C k ( M ) stand for the Banach space of functions on M with k continuous derivatives.The norm on f ∈ C k is given by(3) sup x ∈ M, ≤ i ≤ k |∇ i f ( x ) | Recall the following (see [11] for a proof) fundamental theorems:
Theorem 3. If pk < d , we have the embedding (4) L pk → L dpd − kp If pk > d , we have the embedding (5) L pk → C More generally, if p ( k − m ) > d , we have the embedding (6) L pk → C m We have the multiplication map L p · L p ′ → L pp ′ p + p ′ as well as the following theorem: Theorem 4. If pk > d , the spaces L pk form a Banach algebra under the operation ofpointwise multiplication. At times, it is useful to have a definition of Sobolev spaces for negative k : efinition 6. Let k be a negative integer and let /p + 1 /q = 1 . We set L pk ( M ) = ( L q − k ( M )) ∗ where ( L q − k ( M )) ∗ denotes the dual of L q − k ( M ) . In the case p = 2, a spectral definition of Sobolev norms is useful. Let ∆ = d ∗ d bethe scalar Laplacian for functions on M and let φ λ be an orthonormal eigenbasis of∆. For any smooth f , we have the decomposition f = X λ c λ φ λ with c λ = h φ λ , f i L . We may define the L k -norm by setting(7) || f || L k = X λ | c λ | ( | λ | + 1) k/ Standard elliptic estimates imply that this definition yields a norm equivalent to (2)in the case p = 2 (see [9]). We may therefore alternatively define L k ( M ) as the com-pletion of smooth functions with respect to this norm. One advantage of the spectraldefinition is that it immediately extends to all k ∈ R .Let us now turn to the case of a manifold with boundary. Let M + be a compact,oriented, Riemannian manifold with boundary ∂M + . We will assume that near theboundary, M + is isometric to ∂M + × [0 , M − denote a copy of M + with theopposite orientation. We let M = M + ∪ ∂M + M − be the double manifold formed by gluing two copies M + and M − along the boundary.One can easily extend the definition of Sobolev spaces to the case of a manifoldwith boundary. Indeed, for k ∈ { , , . . . } , one may define L pk ( M + ) by completingthe space of smooth functions on M + with respect to the norm (2). We will makeuse of the following basic extension lemma: Lemma 1.
There exists a continuous linear extension map E : L pk ( M + ) → L pk ( M ) such that E ( f ) | M + = f for all f ∈ L pk ( M + ) .Proof. The proof is contained in [9] and we give a sketch of the construction for lateruse. For this, we construct E : L pl ( M + ) → L pl ( M ) or all l ≤ k as follows. First, we locally identify M + with ( x, y ) ∈ R d − × [0 , x are the ∂M + coordinates and y is the normal coordinate. Fix some choiceof coefficients a j . Let E ( f ) be f ( x, y ) for y ≥ k +1 X j =1 a j f ( x, − jy )for y <
0. As explained in [9], there exists a unique choice of coefficients a j so thatall derivatives up to order k match up at the boundary for all f . (cid:3) In view of the previous lemma, we may alternatively define L pk ( M + ) as(8) L pk ( M + ) = L pk ( M ) /L pk ( M ) M − where L pk ( M ) M − ⊂ L pk ( M ) consists of elements with support in M − . This definitionallows one to extend L pk ( M + ) to all real k .Let R : C ∞ ( M + ) → C ∞ ( ∂M + )denote the restriction map. We have the following trace theorem (see [9]): Theorem 5. If k > / , R extends to a continuous surjective map (9) R : L k ( M + ) → L k − / ( ∂M + ) For any p > , there is a continuous restriction map (10) R : L p ( M + ) → L p ( ∂M + )2.2. Some Nonlinear Estimates.
In dealing with nonlinear estimates, it will beconvenient for us to introduce an alternative notation for Sobolev spaces. To this end,we will often write L /q ( M ) k instead of L p ( M ) k where pq = 1. Note the embedding(11) L /q → L /r for q ≤ r . The multiplication lemma for Sobolev spaces may now be expressed as(12) L /q · L /q → L / ( q + q ) as long as q + q ≤
1, while the basic embedding theorem 3 is now expressed as(13) L /q → L / ( q − d − ) for q − /d > d = 4and fix 1 / < p < /
2. We have the embeddings(14) L /p · L /p → L / ( p − / · L /p → L / (2 p − / (15) L /p · L /p → L / ( p − / · L / ( p − / → L / (2 p − / L / (2 p − / → L / (2 p − / In addition, we claim that(17) L /p · L / (2 p − / → L / (2 p − / Indeed, by applying derivatives, we obtain L /p · L / (2 p − / → L / (3 p − / and L /p · L / (2 p − / → L / (3 p − / Finally, observe that 2 p − / > p − / / > p . Therefore, weobtain the embedding L / (3 p − / → L / (2 p − / and thus (17) as desired.2.3. Sobolev Spaces for Banach Valued Functions.
Let B be a separable Ba-nach space and let M be a closed Riemannian manifold of dimension d . In practice,we will always assume that B = L ql (Σ) × R N for some compact manifold Σ. Muchof the Sobolev space theory discussed in the previous sections directly generalizes tothe case of f : M → B Below, we explain the bare minimum we will use in this work. Since many of theproofs are identical to their finite dimensional analogues, we will present a very con-densed account.Let p > k a nonnegative integer. On smooth maps f : M → B , we definethe L pk -norm by k X i =0 ( Z M |∇ i f | p ) /p Let L pk ( M ; B ) denote the completion of the space of smooth functions with respectto the L pk -norm. We have the following basic approximation lemma: Lemma 2.
Let Σ be a compact Riemannian manifold and B = L ql (Σ) . The space C ∞ ( M × Σ) is dense in L pk ( M ; B ) .Proof. Using a partition of unity, it suffices to prove the result when M = T d , where T d is a d -dimensional torus with coordinates x i . Given f ∈ L pk ( T d ; B ), we mayassume that f ∈ C ∞ ( T d ; B ) since such functions are dense in L pk ( T d ; B ). The Fourierinversion theorem (valid for Banach valued maps) implies that f = X λ c λ e λ here c λ ∈ B and e λ = e ix λ + ...ix d λ d is an eigenfunction of ∆ T d = − P di =1 ∂ x i . We may approximate f in the C k -norm bya finite sum | λ | Lemma 3. We have L p ( M × Σ) = L p ( M ; L p (Σ)) and L pk ( M ; L p (Σ)) = L p (Σ; L pk ( M )) .Proof. The corresponding norms agree on C ∞ ( M × Σ). The previous lemma impliesthat C ∞ ( M × Σ) is dense in all the spaces considered. Therefore, we get the desiredconclusion by taking completions. (cid:3) Remark. As an application of lemma 3, note that the equality L pk ( M ; L p (Σ)) = L p (Σ; L pk ( M ))allows us to define L pk ( M ; L p (Σ)) for negative k by taking the right hand side as thedefinition of L pk ( M ; L p (Σ)). Lemma 4. We have L pk ( M × Σ) = L pk ( M ; L p (Σ)) ∩ L p ( M ; L pk (Σ)) .Proof. First of all, we note that L pk ( M × Σ) ⊂ L pk ( M ; L p (Σ)) ∩ L p ( M ; L pk (Σ))since C ∞ ( M × Σ) are dense in all the spaces considered and the norm on L pk ( M × Σ)controls the norm on L pk ( M ; L p (Σ)) and L p ( M ; L pk (Σ)). To establish the claim we firstshow that f ∈ L pk ( M ; L p (Σ)) ∩ L p ( M ; L pk (Σ)) can be simultaneously approximated bya single smooth function. For this, we assume that M = T d as in the previous proof.Let g ǫ : T d → R be a smooth mollification of the Dirac delta function (see [9] for thedetails). Let f ǫ = f ∗ g ǫ be the convolution of f with g ǫ . By taking ǫ sufficiently small, we may replace f by f ǫ ∈ C ∞ ( M ; L p (Σ)) ∩ C ∞ ( M ; L pk (Σ)) which approximates f in both normssimultaneously. Now, we develop f ǫ in a Fourier series f ǫ = X λ c λ e λ As above, we approximate f ǫ by a finite truncated series f ǫ = P | λ |≤ Nλ c ′ λ e λ thatapproximates f arbitrary closely in both the L pk ( M ; L p (Σ))-norm as well as the L p ( M ; L pk (Σ))-norm. o prove the claim, we argue as follows. For k = 1, the norms on the two sidesagree. For k = 2, let f ∈ L pk ( M ; L p (Σ)) ∩ L p ( M ; L pk (Σ)). We apply the Laplacian∆ M × Σ f = ∆ M f + ∆ Σ f ∈ L p ( M × Σ)By elliptic regularity, (see [11] and the following section) we have f ∈ L p ( M × Σ).To prove the general case we need to construct an appropriate analogue of ∆. Foreven k , we may take ∆ k/ M + ∆ k/ . This is an elliptic differential operator of order k/ f ∈ L pk ( M × Σ). For odd k , one has a variant of thisargument using Dirac operators. (cid:3) Let M + be compact Riemannian manifold with boundary ∂M + . We will assumefor simplicity that M + is isometric to a product [0 , × ∂M + near the boundary. Wewould like to generalize the previous results to this case. Lemma 5. L pk ( M + × Σ) = L pk (Σ; L p ( M + )) ∩ L p (Σ; L pk ( M + )) .Proof. Given f ∈ L pk (Σ; L p ( M + )) ∩ L p (Σ; L pk ( M + )), we use the extension lemma 1 toconstruct E ( f ) ∈ L pk (Σ; L p ( M )) ∩ L p (Σ; L pk ( M ))By lemma 4, we have E ( f ) ∈ L pk ( M × Σ). Restricting to M + × Σ gives the desiredresult. (cid:3) Linear Elliptic Estimates Elliptic Operators on Closed Manifolds. We now briefly review regularitytheory for elliptic differential operators on closed manifolds. The proofs of all theresults can be found in [11].Let M be a closed Riemannian manifold and let D : Γ( E ) → Γ( F )be a differential operator acting on sections of Hermitian vector bundles over M . Weassume that D has smooth coefficients. Let L pk (Γ( E )) stand for the completion of thespace of smooth sections of E with respect to the norm from equation 2. If D hasorder m , it extends to a continuous map L pk (Γ( E )) → L pk − m (Γ( F )) for all real k and p > 1. As before, we will often drop Γ( E ) from the notation.Let us assume that D is elliptic. By definition, this means that the principal symbolof D is invertible (see [9]). We have the following fundamental result: heorem 6. For f ∈ L pk (Γ( E )) , we have || f || L pk ≤ C p,k ( || Df || L pk − m + || f || L pk − ) Furthermore, if f ∈ L pk and D ( f ) ∈ L p ′ k ′ we have that f ∈ L p ′ k ′ + m . Here are some examples of elliptic operators that occur in this work: • Given a Hermitian connection ∇ on a vector bundle, we may form the con-nection Laplacian ∇ ∗ ◦ ∇ . This is an elliptic operator of order 2. • Given a Hermitian connection ∇ on a vector bundle E , we may form the firstorder operator d ∇ + d ∗∇ acting on Λ ∗ ( E ) = Λ ∗ ( M ) ⊗ Γ( E ). • Given a holomorphic vector bundle E on a Riemann surface, we have theoperator ∂ : Γ( E ) → Γ( E ⊗ K − )where K − is the anticanonical bundle. • If dim ( M ) = 4, we have the first order elliptic operator (see [3] for a generaldiscussion) d ∗∇ + d + ∇ : Λ ( E ) → Λ ( E ) ⊕ Λ + ( E )Here Λ + is the bundle of self-dual 2-forms.3.2. Dirichlet and Neumann Problem. We discuss the regularity theory for theDirichlet and Neumann boundary value problems with special emphasis on weak so-lutions. Our goal is to explain how to reduce the various regularity statements to theinterior cases. There are many alternative treatments of this material (see [4]).Let M + be compact Riemannian manifold with boundary ∂M + and double M = M − ∪ ∂M + M + We will focus on the L pk -regularity theory for the case k ≤ p > Definition 7. Let f ∈ L p ( M + ) and g ∈ L ( M + ) . We say that f is a weak solution to (18) ∆ f = g and f | ∂M + = 0 if we have (19) h f, ∆ h i M + = h g, h i M + for all smooth h ∈ C ∞ ( M + ) that vanish on ∂M + . Note that if f ∈ L p ( M + ), this condition coincides with the usual definition∆ f = g and f | ∂M + = 0This follows directly from Green’s formula h ∆ u, v i M + − h u, ∆ v i M + = Z ∂M + u∂ ν v − v∂ ν u where ∂ ν is the outward normal derivative at the boundary. Here is the basic regularityresult: Lemma 6. Let f ∈ L p be a weak solution to ∆ f = g , f | ∂M + = 0 . If g ∈ L p , then f ∈ L p .Proof. The basic idea it to extend f and g to the double M and then apply ellipticregularity for the closed manifold M . We extend f to ˜ f ∈ L p ( M ) by taking − f onthe M − piece of M . Similarly, we extend g to ˜ g ∈ L p ( M ) by taking − g on M − . Weneed to show that h ˜ f , ∆ h i M = h ˜ g, h i M for all smooth h on M . Given such a test function h , decompose h as h = h s + h a where h s is symmetric with respect to the reflection across ∂M + and h a is antisym-metric. Since ˜ g and ˜ f are antisymmetric, we have h ˜ g, h s i M = h ˜ f , h s i M = 0Thus, we may assume h = h a . In particular, we may assume that h vanishes on ∂M and h ˜ g, h i M = 2 h ˜ g, h i M + while h ˜ f , ∆ h i M = 2 h ˜ f , ∆ h i M + By hypothesis, these two integrals are equal. Now, we may apply theorem 6 to deducethat ˜ f ∈ L p ( M ) and thus, by restriction, f ∈ L p ( M + ). (cid:3) We now state the regularity results for higher norms: Lemma 7. Let k ≥ , p > . Let f ∈ L p be a weak solution to ∆ f = g , f | ∂M + = 0 .If g ∈ L pk , then f ∈ L pk +2 .Proof. This follows by induction from the previous lemma as explained for instancein [9]. (cid:3) We turn now to the corresponding Neumann boundary value problem. Let ∂ ν f beoutward normal derivative on ∂M + . Definition 8. Let f ∈ L p ( M + ) and g ∈ L ( M + ) . We say that f is a weak solutionto (21) ∆ f = g and ∂ ν f = 0 f we have (22) h f, ∆ h i = h g, h i for all smooth h ∈ C ∞ ( M + ) with ∂ ν h = 0 . Here is the corresponding regularity result: Lemma 8. Let f ∈ L p be a weak solution to ∆ f = g , ∂ ν f = 0 . If g ∈ L p , then f ∈ L p .Proof. The argument is virtually identical to lemma 6. The main difference is thatnow one uses the symmetric extension of f to M instead of the antisymmetric exten-sion of lemma 6. (cid:3) We now state the regularity results for higher norms: Lemma 9. Let k ≥ , p > . Let f ∈ L p be a weak solution to ∆ f = g , ∂ ν f = 0 . If g ∈ L pk , then f ∈ L pk +2 .Proof. This follows by induction from the previous lemma as explained for instancein [9]. (cid:3) We consider now the inhomogeneous case of these equations. We focus on theNeumann case as is it less standard. Here is the fundamental result of Nirenburg thatwe will use (see [4] for a proof): Theorem 7. Let p > and k ≥ . There exists a continuous map (23) T : L pk ( M + ) → L pk +1 ( M + ) such that for every f ∈ L pk ( M + ) we have (24) ∂ ν T ( f ) = f | ∂M + Proof. This theorem is essentially contained in [4] and thus we restrict ourselves to abrief sketch. Let us first consider the local construction. Let H = { ( s, t ) ∈ R | s ≥ } be the half space and consider f ∈ L p ( H ) with support on the standard unit disk D . Let K ( s, t ) be defined by K ( s, t ) = ln( s + t )2 π Let g ( s, t ) = − Z R K ( s, t − τ ) f | R ( τ ) dτ By construction (see [4]), ∂ s g (0 , t ) = − f | R nd || g || L p ≤ C || f || L p Now, we will obtain a compactly supported modification of g using a bump function.Let ρ : R → R be a bump with support in D such that ρ = 1 on D . Let ρ ( s, t ) = ρ ( s ) ρ ( t )By construction, ∂ s ρ = 0 on ∂H . The compactly supported ρ g has ∂ s ρ g = − f on ∂H as desired. To obtain the global operator T , we proceed as above using apartition of unity near ∂M + . (cid:3) Definition 9. Let f ∈ L p ( M + ) , r ∈ L p ( M + ) and g ∈ L ( M + ) . We say that f is aweak solution to (25) ∆ f = g and ∂ ν f = r | ∂M + if we have (26) h f, ∆ h i M + = h g, h i M + + h r, h i ∂M + for all smooth h ∈ C ∞ ( M + ) with ∂ ν h = 0 . We summarize the results for the inhomogenous Neumann problem with the fol-lowing: Lemma 10. Let f ∈ L p be a weak solution to ∆ f = g , ∂ ν f = r | ∂M + . If g ∈ L p , r ∈ L p then f ∈ L p .Proof. By theorem 7, we may take u ∈ L p ( M + ) with ∂ ν u = f | ∂M + . Let f ′ = f − u .By construction, f ′ satisfies the homogeneous (weak) Neumann problem:(27) ∆ f ′ = g − ∆ u and ∂ ν f ′ = 0Since the L p -norm of u is controlled by the L p -norm of r , we may apply lemma 8 todeduce f ∈ L p as desired. (cid:3) We also have a version of this result for higher Sobolev norms: Lemma 11. Assume k ≥ . Let f ∈ L pk be a solution to ∆ f = g , ∂ ν f = r | ∂M + . If g ∈ L pk , r ∈ L pk +1 then f ∈ L p k . The proof is a straightforward inductive argument using lemma 10 as the base case.For details, see [9]. .3. Elliptic Theory for Banach Space Valued Functions. We now extend theregularity results discussed in the previous sections to the setting of Banach valuedmaps. As before, let M be a closed manifold of dimension d and let B = L p (Σ).Much of the regularity theory for elliptic operators carries over to the setting ofBanach valued maps. Here is the basic result: Lemma 12. For k ≥ , we have an isomorphism ∆ + 1 : L pk ( M ; B ) → L pk − ( M ; B ) In particular, || f || L pk ≤ C p.k ( || ∆ f || L pk − + || f || L pk − ) Proof. Note that ∆ + 1 : L pk ( M ) → L pk − ( M )is an isomorphism for any k . Since by lemma 3 L pk ( M ; B ) = L p (Σ; L pk ( M ))we have the induced isomorphism∆ + 1 : L p (Σ; L pk ( M )) → L p (Σ; L pk − ( M )) (cid:3) Along with the elliptic estimate we have a regularity result. Given f, g ∈ L p ( M ; B ),we say that f is a weak solution to ∆ f = g if for all h ∈ C ∞ ( M × Σ), we have h f, ∆ h i M = h g, h i M Note that the pairings are well defined since f, g ∈ L p ( M × Σ). Lemma 13. Let k ≥ . Assume f ∈ L p ( M ; B ) and ∆ f ∈ L pk ( M ; B ) . We have f ∈ L pk +2 ( M ; B ) .Proof. We address the case k = 0 as the other cases are similar. By the previouslemma, we may take u ∈ L p ( M ; B ) such that (∆ + 1) u = (∆ + 1) f ∈ L p . Thus, byconsidering f − u we may assume that(∆ + 1) f = 0in the weak sense. In particular, this implies that h f, (∆ + 1) h i M = 0for all smooth h . Since (∆ + 1) h is dense in L p ( M ; B ) by lemma 3 we conclude that f = 0 as desired. (cid:3) e now discuss how the elliptic estimates for the Dirichlet and Neumann problemcarry over to Banach space valued functions. Just as in the closed case, the proofs ofthe various regularity results can be reduced to their finite dimensional counterparts. Definition 10. Let f ∈ L p ( M + ; B ) and g ∈ L ( M + ; B ) . We say that f is a weaksolution to (28) ∆ f = g and f | ∂M + = 0 if we have (29) h f, ∆ h i M + = h g, h i M + for all smooth h ∈ C ∞ ( M + × Σ) that vanish on ∂M + × Σ . Here is the corresponding regularity result: Lemma 14. Let f ∈ L p be a weak solution to ∆ f = g , f | ∂M + = 0 . If g ∈ L p , then f ∈ L p .Proof. The reduction to the interior case is identical to the proof of lemma 6, thistime using lemma 13. (cid:3) We now state the regularity results for higher norms: Lemma 15. Let k ≥ , p > . Let f ∈ L p be a weak solution to ∆ f = g , f | ∂M + = 0 .If g ∈ L pk , then f ∈ L pk +2 .Proof. This follows by induction from the previous lemma as in case of lemma 7. (cid:3) We turn now to the corresponding Neumann boundary value problem. Let ∂ ν f beoutward normal derivative on ∂M + . Definition 11. Let f ∈ L p ( M + ; B ) and g ∈ L ( M + ; B ) . We say that f is a weaksolution to (30) ∆ f = g and ∂ ν f = 0 if we have (31) h f, ∆ h i M + = h g, h i M + for all smooth h ∈ C ∞ ( M + × Σ) with ∂ ν h = 0 . Here is the corresponding regularity result: Lemma 16. Let f ∈ L p be a weak solution to ∆ f = g , ∂ ν f = 0 . If g ∈ L p , then f ∈ L p .Proof. The argument is similar to lemma 14. The main difference is that now oneuses the symmetric extension of f to M instead of the antisymmetric extension inlemma 14. (cid:3) e now state the regularity results for higher norms: Lemma 17. Let k ≥ , p > . Let f ∈ L p be a weak solution to ∆ f = g , ∂ ν f = 0 .If g ∈ L pk , then f ∈ L pk +2 .Proof. This follows by induction from the previous lemma as in case of lemma 9. (cid:3) We consider now the inhomogeneous case of these equations. As before, we focuson the Neumann case. Lemma 18. Let p > and k ≥ . There exists a continuous map (32) T : L pk ( M + ; B ) → L pk +1 ( M + ; B ) such that for every f ∈ L pk ( M + ; B ) we have (33) ∂ ν T ( f ) = f | ∂M + Proof. Since L pk ( M + ; L p (Σ)) = L p (Σ; L pk ( M + )), we define T : L p (Σ; L pk ( M + )) → L p (Σ; L pk +1 ( M + ))using theorem 7. (cid:3) Definition 12. Let f ∈ L p ( M + ; B ) , r ∈ L p ( M + ; B ) and g ∈ L ( M + ; B ) . We saythat f is a weak solution to (34) ∆ f = g and ∂ ν f = r | ∂M + if we have (35) h f, ∆ h i M + = h g, h i M + + h r, h i ∂M + for all smooth h ∈ C ∞ ( M + × Σ) with ∂ ν h = 0 . We summarize the results for the inhomogenous Neumann problem with the fol-lowing: Lemma 19. Assume k ≥ . Let f ∈ L pk be a solution to ∆ f = g , ∂ ν f = r | ∂M + . If g ∈ L pk , r ∈ L pk +1 then f ∈ L p k .Proof. The proof is identical to the case when B is finite dimensional. (cid:3) Finally, we point out that the previous lemmas have an extension to the case B = L p (Σ) ⊕ R N . In this case, we separate the finite dimensional part and treat itusing the methods discussed above.We now turn to regularity results for first order elliptic operators. Consider somefirst order elliptic differential operator D : Γ( E ) → Γ( F ) cting on sections of bundles over M . If we let B p = L p (Σ), we get an inducedoperator D : L pk ( M ; B p ⊗ E ) → L pk − ( M ; B p ⊗ F )for all k ≥ 0. We have: Lemma 20. Let k ≥ . Given f ∈ L pk ( M ; B p ⊗ E ) , we have || f || L pk ≤ C ( || Df || L pk − + || f || L pk − ) Assume, Df ∈ L qk − ( M ; B q ) and f ∈ L qk − ( M ; B q ) with q > p . We have f ∈ L qk ( M ; B q ) .Proof. By considering D ′ = D ⊕ D ∗ , we may as well assume that D is self-adjoint.Furthermore, for a generic choice of c ∈ R , D + c is invertible as a map L pk → L pk − .As before, we identify L pk ( M ; B p ⊗ E ) = L p (Σ; L pk (Γ( E ))) and obtain the isomorphism D + c : L pk ( M ; B p ⊗ E ) → L pk − ( M ; B p ⊗ E )The rest of the proof is identical to the proof of lemma 12. (cid:3) Remark. Let B p = L p ( N ) × R n and let B p = B p ⊕ B p with complex structure J that maps ( a, b ) to ( − b, a ). We obtain the corresponding ∂ -operator L pk ( T ; B p ) → L pk − ( T ; B p )for all k ≥ 1. This example is a particular case of the previous construction wherewe take D = ∂ and E = C n . Note that ∂ + 1 is an isomorphism on L pk ( T ; B p ). Remark. Let D R ⊂ C be the closed disk of radius R . For any k ≥ 1, let L pk ; L ( D R ; C )be subspace of L pk ( D R ; C ) with imaginary values on ∂D R . Consider the operator ∂ : L pk ; L ( D R ; C ) → L pk − ( D R ; C )as discussed in [15]. This operator is surjective with left inverse denoted by T . Notethat B p = B p ⊗ R C and B p ⊗ R ( i R ) is Lagrangian. This implies that the operator ∂ : L pk ; L ( D R ; B p ) → L pk − ( D R ; B p )is surjective with a left inverse induced by T . . The Moduli Space of Projectively Flat Connections on a Surface Basic Construction. Consider a compact Riemann surface Σ with complexstructure j Σ and Kahler metric g Σ . Let E → Σ be a 2-dimensional complex vectorbundle with odd c ( E ) ∈ H (Σ; Z ) ∼ = Z Let V → Σ be the bundle of traceless endomorphisms of E . Note that V has struc-ture group SO (3). We fix once and for all a unitary connection α det on det ( E ). Oncethis choice of α det is made, we have an identification between SO (3)-connections on V and U (2)-connections on E that induce α det on det ( E ).Let G (Σ) E be the group of U (2)-gauge transformations of E that descend to theidentity on det ( E ). G (Σ) has a natural induced action on the connections on V andwe let G (Σ) ⊂ G (Σ) V be its image. We may identify the action of G (Σ) with theaction of SO (3)-gauge transformations on V that lift to gauge transformations of E . Definition 13. A p,k be the affine space of SO (3) -connections on V completed with re-spect to the L pk -norm . A p,k is a space modeled on the space of traceless endomorhismsof V , Ω (Σ; g ) . Definition 14. Let G (Σ) p,k +1 be completion of G (Σ) with respect to the L pk +1 -topology. G (Σ) p,k +1 is a fibre bundle with fibre SO (3) . We will assume that p ( k + 1) > G (Σ) acts on A by g ∗ ( α ) = α + g − d α g Following Atiyah and Bott [2], we note that this action is Hamiltonian with momentmap µ : A p,k → Ω p,k (Σ; g )given by µ ( α ) = ∗ F α , where F α is the curvature of α . Lemma 21. µ is a smooth map L pk → L pk − for ( k + 1) p > . Furthermore, ∈ Ω (Σ; g ) is a regular value of µ .Proof. Let us assume that k = 0, as the other cases are easier. We examine thenonlinear part of the moment map that sends α to α ∧ α . Let p ∗ + p = 1. Indimension 2, we have the embedding L p ∗ → L p ∗ − p ∗ and L p · L p ⊂ L p/ Therefore, α ∧ α defines an element of L p − by the pairing L p · L p · L p ∗ − p ∗ ⊂ L ince 2 p + 2 − p ∗ p ∗ = 1 p + 12 < µ ( α ) = 0 then α is flat and thus must be irreducible. This implies that Dµ α = ∗ d α is surjective. (cid:3) Let C ⊂ A be the set of projectively flat connections. Definition 15. Let M = µ − (0) / G (Σ) = A // G (Σ)In fact, M is a compact Kahler manifold (see [2]). M has a concrete descriptionin terms of representations of π (Σ). Pick a point p ∈ Σ. The space M is the spaceof representations π (Σ − p ) → SU (2) that have holonomy − I around p , moduloconjugation by SU (2). If we pick a standard homology basis { α i } gi =1 , { β i } gi =1 , wemay identify M with the space(36) { g i ∈ SU (2) , h i ∈ SU (2) | Π gi =1 [ g i , h i ] = − I } /SU (2)We now describe a convenient local parametrization for C and M . Fix a flat connec-tion α . Consider the map(37) ˜ F : A p,k → Ω p,k − (Σ; g ) ⊕ Ω p,k − (Σ; g ) ⊕ H α (Σ)where ˜ F = µ ⊕ d ∗ α ⊕ Π and Π is the projection to the finite dimensional space ofharmonic ( d ∗ α + d α )-forms. This map is a diffeomorphism near α and gives us a localparametrization of M and C . The linearization of ˜ F at a point α ∈ A gives us themap D ˜ F α : Ω p,k (Σ; g ) → Ω p,k − (Σ; g ) ⊕ Ω p,k − (Σ; g ) ⊕ H α (Σ)An important technical point that will be useful in establishing regularity is that foreach α ∈ L p , D ˜ F α extends to a continuous map L q → L q − for all q ≥ p ∗ . This is immediate for Π and d ∗ and thus we need only treat thecomponent given by µ . Since Dµ a ( v ) = d v + 2[ α, v ]the continuity of the extension is a consequence of the fact that L p · L q → L q − To justify this embedding we argue as follows. By definition, L q − = ( L q ∗ ) ∗ where1 /q + 1 / ( q ∗ ) = 1. Thus, to justify the embedding we need to establish that L p · L q · L q ∗ → L his follows by direct computation. Alternatively, recasting this claim in the notationof section 2 . 2, we must establish L /p · L /q · L / (1 − /q )1 → L where p = 1 /p and q = 1 /q . First assume q > 2. We have L / (1 − q )1 → L / (1 − q − / and L /p · L /q · L / (1 − q )1 → L /r where r = p + q + 1 − q − / p + 1 / < p > / 2. Now take q < 2. In thiscase L q ∗ ⊂ L ∞ and thus L /p · L /q ⊂ L / ( p + q ) ⊂ L as long as p + q ≤ 1. The case q = 2 is similar.We also note that the map D ˜ F : Ω p, (Σ) → End (Ω q, (Σ; g ) q, , Ω q, − (Σ; g ) ⊕ Ω q, − (Σ; g ) ⊕ H α (Σ))is a smooth map of α . Indeed, D ˜ F is linear and continuous in α and therefore smooth.To abstract the situation, let V p = L p (Σ) be the space of L p -sections of some bundleover Σ. We say that a map T : V p → End ( V p , V p )is compatible with the underlying L q -structure if it extends to a smooth map T : V p → End ( V q , V q )for p ∗ ≤ q .4.2. The Canonical Lagrangian. Given a symplectic manifold ( M, ω ) with a Hamil-tonian group action G and corresponding moment map µ : M → Lie ( G )one may form the symplectic reduction at 0 by M//G = µ − (0) /G Provided G acts freely on µ − (0), ( M//G, ω M//G ) inherits a symplectic structure from M . Let ( M//G ) − denote the symplectic manifold with the opposite form. There isa canonical Lagrangian L = µ − (0) ⊂ ( M//G ) − × M defined as the set of pairs ([ m ] , m ). Here, [ m ] = mG denotes the orbit of m . Thisgeneral construction applies to the case of interest where M = A and G = G (Σ). Definition 16. Let L p,k ⊂ M × A p,k be the set of pairs ([ α ] , α ) where α is a flatconnection on Σ . he goal of the present section is to construct a convenient choice of local chartsfor L .Let U ⊂ H α be an open ball around the origin. If U is sufficiently small, we have alocal diffeomorphism(38) f : U → M around a point [ α ] ∈ M . We will let j denote the induced complex structure on U .Let C ⊂ A denote the submanifold of flat connections. We have that ˜ F from equation(37) gives us a local identification of C with an small open ball V p,k ⊂ Ω p,k − (Σ) ⊕ H α Let J = ( j , ∗ Σ ) denote the product complex structure on U × A p,k . If we restrict theinverse of F to V p,k , we obtain a local embedding of the canonical Lagrangian G : V p,k → U × A p,k where G ( v ) = ([ F − ( v )] , F − ( v )). We may extend G to obtain a local chart for U × A p,k by the map H : V p,k ⊕ V p,k → U × A p,k that sends ( u, v ) G ( u ) + J ( G ( v ))This provides a local identification of L with ( u, 0) and that along L the inducedcomplex structure sends ( u, v ) to ( − v, u ). As in the previous section, DH preservesthe L q -structure on V p,k ⊕ V p,k in the case k = 0 and extends to a smooth mappingbetween these spaces. 5. A Priori Estimates ASD Equation/J-Curve Equations. Let us setup some basic conventions.Let X be a smooth oriented 4-manifold with metric g X . In this work we will beinterested in X ⊂ C × Σ with the product metric. We will use ( x, y ) for the localcoordinates on Σ and ( s, t ) as coordinates on C . We have the Hodge star operator indimension 4: ∗ ( dxdy ) = dsdt ∗ ( αdt ) = ( ∗ α ) ds ∗ ( αds ) = − ( ∗ α ) dt If A is an SO (3)-connection, we have the gauge group action:(39) g ∗ ( ∇ A ) s = g − ∇ A ( gs ) = g − ( d + A ) gs ) = ds + g − dgs + g − Ags = ∇ A s + g − ∇ A g n C × Σ, we can decompose our connection A as A = α + φds + ψdt and its curvature as F A = F α − ∂ t αdt − ∂ s αds + ( d φ + [ α, φ ]) ds + ( d ψ + [ α, ψ ]) dt + ( ∂ s ψ − ∂ t φ + [ φ, ψ ]) dsdt (40)The anti-self-duality (ASD) equation(41) F A + ∗ F A = 0becomes the pair of equations ∂ s α − d α φ + ∗ ( ∂ t α − d α ψ ) =0 ∗ F α + ∂ s ψ − ∂ t φ + [ φ, ψ ] =0(42)In general, the energy of a connection A is defined as(43) 12 Z X | F A | dµ X where dµ X is the volume element associated to g X . On a closed 4-manifold X , thesecond Chern class is given by the formula(44) c ( P ) = 18 π Z X tr ( F A ) = 18 π Z X tr (( F + A ) )+ tr (( F − A ) ) == 18 π Z X ( | F − A | −| F + A | ) dµ X where we use the convention that | D | = tr ( D ∗ D ) = − tr ( D ) for any skew-hermitianendomorphism D . Thus, for an ASD connection A , we have(45) c ( P ) = 18 π Z X | F A | dµ X Let M = M (Σ) be the representation variety as in section 4 and let let D be theopen unit disk. A holomorphic curve u : D → M with C small image may be liftedto a map α : D → A satisfying ∂ s α − d α φ + ∗ ( ∂ t α − d α ψ ) =0 F α =0 d ∗ α ( α − α ) =0(46)where α = α (0). This is a consequence of the inverse function theorem applied tothe map from equation (38). It is instructive to compare equation (46) to (42). .2. Matching Boundary Conditions. Let B R ( p ) ⊂ C be the closed disk of radius R centered at p and let H + = { ( s, t ) ∈ C | s ≥ } be the positive half-plane. We define D R ( p ) as H + ∩ B R ( p ). Perhaps it is more naturalto use D + R as notation. However, since we will work mostly on the positive half planewe drop the + for simplicity. Let D − R ( p ) be the reflection of this disk in the t -axis.In general, ∂D R = I R ∪ S R where I R = { (0 , t ) ∈ D R } and S R = { ( s, t ) ∈ D R ( p ) | s + t = R } We will often drop p from the notation when the p does not change in a particulardiscussion. The interior of a disk D R is the set of points with s + t < R and willbe denoted by ˚ D R .Consider a holomorphic map u : D − R → M and an ASD connection A on D R × Σ. For each t ∈ I R , we may restrict A to the slice(0 , t ) × Σ. This gives us a map R A : I R → A (Σ) Definition 17. The pair ( u, A ) is said to be matched if at each t ∈ I R , u (0 , t ) =[ R A ( t )] where [ R A ( t )] denotes the gauge orbit of R A ( t ) . Note that this is precisely the condition that ( u (0 , t ) , R A ( t )) ∈ L for each t . If A was a holomorphic curve, this would amount to a Lagrangian boundary condition forthe pair (˜ u, A ) : D R → A (Σ)where ˜ u ( s, t ) = u ( − s, t ). For convenience, we will refer to the pair ( u, A ) as definedon D R × Σ.5.3. Statement of the Result. Let f ± be the functions defined by f + ( s, t, x, y ) = F A ( s, t, x, y )and f − ( s, t, x, y ) = | du ( − s, t ) | We will also make use of e : D R → R defined by e ( s, t ) = e + ( s, t ) + e − ( s, t ) where e + ( s, t ) = 12 Z Σ | f + ( s, t, x, y ) | dµ Σ nd e − ( s, t ) = 12 f − ( s, t ) with ∂ s e = Z Σ h f + , ∇ s f + i + f − ∂ s f − The estimates below will keep track of the radius R . We will always assume that allthe constants do not depend on R or the choice of functions f ± . The following resultis key in our proof of compactness for matched pairs. Theorem 8. There exists ~ , C > with the following property. Let ( u, A ) be amatched pair on some D R × Σ . If E ( u, A ) < ~ then e ( p ) ≤ C E ( u, A ) R − The rest of the section is devoted to the proof of this theorem.5.4. Weitzenb¨ock Formulae. Given an SO (3)-connection on a 4-manifold X , wehave the Weitzenb¨ock formula ∇ ∗ A ∇ A F A + { F A , R X } + { F A , F A } = ( d ∗ A d A + d A d ∗ A ) F A where the brackets denote some pointwise multiplication and the term R X dependsonly on the metric of X . See [13] for a detailed discussion. What is important forour purposes is that the order zero terms are at most quadratic in F A . If F A is ASD,we have that in particular F A satisfies the Yang-Mills equation d ∗ A F A = 0and therefore(47) ∇ ∗ A ∇ A F A = −{ F A R X } − { F A , F A } An application of the Weitzenb¨ock formula (see [14] for the holomorphic curve case)leads to pointwise estimates:(48) ∆ | f + | ≤ C ′ ( | f + | + | f + | ) − |∇ A F A | ≤ C ′ ( | f + | + | f + | )(49) ∆ f − = ∆ f − ≤ C ′ ( f − + f − ) − | du | ≤ C ′ ( f − + f − )For a given nonnegative function g , the relation∆ g = 2 g ∆ g − |∇ g | ≤ g ∆ g leads to(50) ∆ | f + | ≤ C ′ ( | f + | + | f + | )(51) ∆ f − = ∆ f − ≤ C ′ ( f − + f − ) ntegration of (48) on Σ gives∆ Z Σ | f + | dµ Σ ≤ C ′ ( Z Σ | f + | dµ Σ + v ( s, t )( Z Σ | f + | dµ Σ ) / )where v ( s, t ) = ( Z Σ | f + | ( s, t ) dµ Σ ) / is obtained from the Cauchy-Schwartz inequality: Z Σ | f + | dµ Σ ≤ ( Z Σ | f + | dµ Σ ) / ( Z Σ | f + | dµ Σ ) / Thus, we have(52) ∆ e ≤ C ′ ( e + e + e / v )5.5. Normal Estimates. So far, we have not used the matching boundary condi-tions and thus the ASD connection and the holomorphic curve do not interact. Inthis section we will demonstrate how the matching boundary conditions lead to anormal estimate for e . In fact, we have the following result: Lemma 22. For each (0 , t ) ∈ D R we have | ∂ s e | = | Z Σ h f + (0 , t ) , ∇ s f + (0 , t ) i + f − (0 , t ) ∂ s f − (0 , t ) | ≤ Ce / (0 , t )The proof of this lemma will occupy the rest of this section. Since our estimate islocal in ( s, t ), we may assume that our disk D R is centered at the origin. The sizeof the radius is not relevant and we can set it to R = 1. We begin by constructinga convenient gauge for A = α + φds + ψdt . First, we fix α (0 , α (0 , 0) is flat and thus, in view of the compactnessof M , we can choose any such α (0 , 0) to have a uniformly bounded C -norm. Wenow construct a particular gauge for A on D × Σ. For this, let us use A to paralleltransport from (0 , × Σ to (0 , t ) × Σ for any t ∈ ( − , D × Σalong the s -direction. In these coordinates, φ = 0 on D × Σ and ψ = 0 on (0 , t ) × Σ.Equation 41 implies that(53) ∂ s α + ∗ ( ∂ t α − d α ψ ) = 0and(54) ∗ F α + ∂ s ψ = 0with F α = 0 on (0 , t ) × Σ. This implies that ∂ s ψ = 0 on (0 , t ) × Σ. Therefore, ∂ s ( d α ψ ) = d α ∂ s ψ + ( ∂ s α ) ψ = 0on (0 , t ) × Σ. Now, the energy (43) is given by | ∂ t α − d α ψ | + | F α | or an ASD-connection. Note that since ∇ s = ∂ s and F α = 0 on (0 , t ) × Σ it followsthat ∂ s Z Σ | f + | dµ Σ = h ∂ s F A , F A i Σ = 2 h ∂ t α, ∂ t ∂ s α i Σ on (0 , t ) × Σ. We may apply ∂ t to (53) on (0 , t ) × Σ to obtain ∗ ∂ t α + ∂ s ∂ t α = 0Therefore, we obtain on (0 , t ) × Σ ∂ s Z Σ | f + | dµ Σ = −h ∂ t α, ∗ ∂ t α i Σ = Z Σ tr ( ∂ t α ∧ ∂ t α )We now establish normal estimates on D − . We let α − (0 , 0) = α + (0 , , t )we lift [ α − (0 , t )] to α − (0 , t ) ∈ A by requiring that d ∗ α − ∂ t α − = 0Now, we extend to D − by lifting along line segments in the s -direction with therequirement that d ∗ α − ( ∂ s α − ) = 0. Thus, for sufficiently small neighborhood of (0 , ∂ s α − − d α − φ ) + ∗ ( ∂ t α − − d α − ψ ) = 0The local existence of such a lift follows from existence of ODE as in [3], Chapter 6.Since F α − = 0 on D − , we have d α − ∂ t α − = d α − ∂ s α − = 0on D − . On (0 , t ), applying d α − to (55) we obtain that d ∗ α − d α − ψ = 0 which impliesthat ψ = 0 since H α − (Σ; g ) = 0. Similarly, applying d ∗ α − on ( s, t ) × Σ we obtain that φ = 0 on ( s, t ) × Σ. Therefore, along (0 , t ) the energy is given by e − = Z Σ | ∂ t α − | while, ∂ s ∂ t α − + ∗ ∂ t α − = 0We have − ∂ s e − = Z Σ h ∂ t α − , ∂ t ∂ s α − − d α − ∂ s ψ i = Z Σ h ∂ t α − , ∂ t ∂ s α − i since d ∗ α ∂ t α − = 0 on (0 , s ). We conclude that − ∂ s e − = Z Σ tr ( ∂ t α − ∧ ∂ t α − )For the rest of this section we focus on connections defined on Σ parametrizedby points in (0 , t ). Thus, we will for instance write α (0 , t ) as α ( t ). Take t ∈ ( − ǫ, ǫ ) ⊂ ( − , t -parameter family of flat connections ˜ α ( t ) on Y = [ − , × Σ with ˜ α ( t ) × Σ = α ( t ) and ˜ α ( t ) − × Σ = α − ( t ). irst, by the matching condition, α ( t ) = g ∗ ( t ) α − ( t ). Taking derivatives, we obtainthat(56) ∂ t α = g − ( ∂ t α − ) g + g − ( d α − ξ ) g with ξ = ∂ t gg − .Now, for each t ∈ ( − ǫ, ǫ ), we define the extension ˜ ξ ( t ) on Y = [ − , × Σ with˜ ξ − × Σ ( t ) = 0, ˜ ξ × Σ ( t ) = ξ ( t ) and || ˜ ξ ( t ) || L / ; Y ≤ C || ξ ( t ) || L ;Σ The existence of such an extension is easy to deduce. Indeed, we have the surjectiverestriction map R : L / ([ − , × Σ) → L ( ∂ [ − , × Σ)Taking a left inverse to R provides such an extension for all t ∈ ( − ǫ, ǫ ). To ex-tend the gauge transformation g ( t ) to Y , we set ˜ g ( t ) to be the unique solution to ∂ t ˜ g ( t ) = ˜ ξ ( t )˜ g ( t ) with ˜ g (0) = 1.By pullback, we can regard α − ( t ) as a connection on Y , Let ˜ α ( t ) = ˜ g ∗ ( t ) α − ( t )for each t ∈ ( − ǫ, ǫ ). By Stokes theorem, Z ∂ ([ − , × Σ) tr ( ∂ t ˜ α ∧ ∂ t ˜ α ) = Z [ − , × Σ dtr ( ∂ t ˜ α ∧ ∂ t ˜ α ) = − Z [ − , × Σ tr ( ∂ t ˜ α ∧ ∂ t ˜ α ∧ ∂ t ˜ α )This follows from the fact that d ˜ α ∂ t ˜ α = 0since ˜ α is flat. Now, applying ∂ t we obtain d ˜ α ∂ t ˜ α = − ∂ t ˜ α ∧ ∂ t ˜ α as desired. We claim that || ∂ t ˜ α (0) || L ; Y ≤ Ce / (0 , 0) for some uniform C . Byequation 56, || ∂ t ˜ α (0) || L ; Y ≤ C ( || ∂ t α − (0) || L ;Σ + || d α − ˜ ξ (0) || L ; l Σ )We have || ∂ t α − (0) || L ≤ Ce / (0 , 0) from the definition and our choice of lift. Now, || d α − ˜ ξ (0) || L ≤ C || d α − ˜ ξ (0) || L / ≤ C ′ || ˜ ξ (0) || L / ≤ C ′′ || ξ (0) || L On the other hand, || ξ (0) || L ≤ Ce / (0 , 0) since we have the bound || ∂ t α (0) || L ;Σ ≤ e / (0) hile H α − (Σ; g ) = 0 and α − (0) can be taken to vary in a precompact set in say the C -norm. Thus, we obtain that ∂ t e (0 , 0) = Z ∂ [ − , × Σ tr ( ∂ t ˜ α (0) ∧ ∂ t ˜ α (0)) ≤ Ce / (0 , Integration By Parts. First, we recall some basic Sobolev embedding andrestriction results in dimension 2: Lemma 23. Consider an L function g on D R ( p ) , vanishing near the boundary givenby S R . For any q ∈ [1 , and large C ′ , we have: || g || L q ,D R ( p ) ≤ C ′ R /q ||∇ g || L ,D R ( p ) || g || L q ,∂D R ( p ) ≤ C ′ R /q ||∇ g || L ,D R ( p ) Proof. First, let R = 1. Since g is assumed to have compact support on D ( p ), wehave by the Sobolev embedding L ⊂ L q (see [9]): || g || L q ,D ( p ) ≤ C ′ ||∇ g || L ,D ( p ) || g || L q ,∂D ( p ) ≤ C ′ ||∇ g || L ,D R ( p ) for some C ′ > 0. Now, if we scale g to g R on a disk of radius R , note that ||∇ g || L isscale invariant. On the other hand || g || L q ,D ( p ) = R − /q || g R || L q ,D R ( p ) and || g || L q ,∂D ( p ) = R − /q || g R || L q ,∂D R ( p ) (cid:3) Given a matched pair on some D R we define the integral energy as E R = Z D R e Here is the key result which allows one to bound the L -norm of f ± in terms ofenergy: Lemma 24. There exists C, C ′ > with the following property. If e ( s, t ) ≤ CR − on D R , we have ||∇ A f + || D R ≤ C ′ R − E R ||∇ f − || D R ≤ C ′ R − E R roof. Let 0 ≤ ρ ≤ B R suchthat ρ = 1 on B R . We can construct ρ once and for all on B and extend to ρ R forall B R by dilation. We have a uniform bound on || ρ R || C and ||∇ ρ R || C ≤ C ′ R − on B R . Consider now f − . We multiply ∇ ∗ ∇ f − = ∆ f − ≤ C ′ ( f − + f − )on both sides by ρ R f − and integrate over D R to obtain h ρ R f − , ∇ ∗ ∇ f − i D R ≤ C ′ Z D R ( ρ R f − + ρ R f − )Now, we move ∇ ∗ to the LHS obtaining h∇ ( ρ R f − ) , ∇ f − i D R ≤ C ′ Z D R ( ρ R f − + ρ R f − ) − Z I R ρ R f − ∂ s f − We move a ρ R to the RHS: h∇ ( ρ R f − ) , ∇ f − i D R − h∇ ( ρ R f − ) , ∇ ( ρ R f − ) i D R = −h∇ ( ρ R ) f − , ∇ ( ρ R ) f − i D R Thus, we obtain after adjusting C ′ : ||∇ ( ρ R f − ) || ≤ C ′ R − || f − || D R + C ′ Z D R ( ρ R f − + ρ R f − ) − Z I R Z Σ ρ R f − ∂ s f − Finally, we use the hypothesis that f − ≤ CR − to obtain the bound ||∇ ( ρ R f − ) || ≤ C ′ R − E R − Z I R Z Σ ρ R f − ∂ s f − after adjusting C ′ once more.Now, we turn to f + . We multiply both sides of equation (47) by ρ R f + and pro-ceeding as we did with f − we obtain ||∇ A ( ρ R f + ) || ≤ C ′ ( R − || f + || + Z D R × Σ f + ( ρ R f + ) ) − Z I R Z Σ h ρ R f + , ∇ s f + i≤ C ′ ( R − || f + || + ( Z D R f ) / ( Z D R × Σ ( ρ R f + ) ) / ) − Z I R Z Σ h ρ R f + , ∇ s f + i We use the Sobolev embedding L ⊂ L in dimension 4 together with Kato’s inequality(57) ∇| f | ≤ |∇ A f | to bound( Z D R f ) / ( Z D R × Σ ( ρ R f + ) ) / ) ≤ C ′ ( Z D R f ) / ( ||∇ A ( ρ R f + ) || + || ρ R f + || ) y assumption, R D R f ≤ C and thus if C < /C ′ we can absorb the term to theLHS to obtain the bound ||∇ A ( ρ R f + ) || ≤ C ′ R − || f + || D R − Z I R Z Σ h ρ R f + , ∇ s f + i = C ′ R − E R − Z I R Z Σ h ρ R f + , ∇ s f + i after adjusting C ′ . Since, by section 5 . 5, we have | Z Σ h f + , ∇ s f + i + + Z Σ f − ∂ s f − | ≤ C ′ e / ≤ C ′ C / R − e we can apply the Sobolev embedding lemmas in dimension 2 to conclude that ρ R | Z (0 ,t ) ∈ D R Z Σ h f + , ∇ s f + i + Z Σ f − ∂ s f − | ≤ C / C ′′ ( ||∇ A ( ρ R f + ) || + ||∇ ( ρ R f − ) || )Taking C sufficiently small, we can absorb the term C / C ′′ ( ||∇ A ( ρ R f + ) || + ||∇ ( ρ R f − ) || )to the left hand side and obtain the desired inequality. (cid:3) Inverting the Laplacian. Let us summarize the situation so far. We have anonnegative function e : D R ⊂ H + → [0 , ∞ )with energy functional E R = Z D R e The function e satisfies the following properties:There exists C > C ′ > e ≤ CR − on D R we have:(1) ( R D R e ) / ≤ C ′ E R R − (2) R ∂D R e ≤ C ′ E R R − (3) ( R ∂D R e ) / ≤ C ′ E R R − / (4) ∆ e ≤ C ′ ( e + e + e / v ) with ( R D R v ) / ≤ C ′ E R R − (5) | ∂ s e (0 , t ) | ≤ C ′ e / (0 , t )Our immediate objective is to use these assumptions to get a pointwise bound on e in terms of energy: Lemma 25. For some C ′′ > , and each D R in the domain, we have e ( p ) ≤ C ′′ E R R − where p is the center of D R The proof of this lemma occupies the rest of this section. First, given C functions f , g , on D R , Green’s formula in dimension 2 gives Z D R f ∆ g − Z D R g ∆ f = Z ∂D R ( f ∂ ν g − g∂ ν f ) n general, ∂D R = I R ∪ S R where I R = { (0 , t ) ∈ D R } and S R = { ( s, t ) ∈ C | s + t = R , s ≥ } If g = ln( R ) − ln(( s − s ) + t ), which is a Green’s function for ∆ at p = ( s , Z ∂D R ( f ∂ ν g − g∂ ν f ) = Z I R ( s fs + t + (ln( R ) − 12 ln( s + t )) ∂ s f ) − R − Z S R f Case 1: D R is the whole disk of radius R .In this case ∂D R = S R . For concreteness we may assume that the disk is centered at p = 0 (one must shift H as well but H does not interact with D R in this case). Wemultiply both sides of (4) by ln( R ) − ln( r ) and use Green’s formula to obtain2 πe ( p ) ≤ C ′ ( Z D R e (ln( R ) − ln( r ))+ Z D R e (ln( R ) − ln( r ))+ Z D R e / v (ln( R ) − ln( r ))+ R − Z ∂D R e )We now bound each of the four terms. For this, it will be convenient to recall thefollowing definite integrals: Z R (ln( r ) − ln( R )) dr = 2 R Z R (ln( r ) − ln( R )) rdr = R / e to bound the terms as follows: Z D R e (ln( R ) − ln( r )) ≤ ( Z D R e ) / (2 π Z R (ln( R ) − ln( r )) rdr ) / ≤ πC ′ E R Z D R e (ln( R ) − ln( r )) ≤ CR − ( Z D R e ) / (2 π Z R (ln( R ) − ln( r )) rdr ) / ≤ πC ′ R − E R Z D R e / v (ln( R ) − ln( r )) ≤ ( C / R − )( Z D R v ) / (2 π Z R (ln( R ) − ln( r )) rdr ) / ≤ πC ′ R − E R R − Z ∂D R e ≤ C ′ R − E R Here we bound e / using the fact that e ≤ CR − on D R . Putting this togetheryields e ( p ) ≤ C ′′ E R R − as desired. ase 2: D R is the half disk with center p = 0. In this case we have an extracontribution to Green’s formula given by Z R − R (ln( r ) − ln( R )) ∂ s edr Since property (4) of e implies that | ∂ s e (0 , t ) | ≤ C ′ e / (0 , t ) we get | Z R − R ∂ t e (ln( r ) − ln( R )) dr | ≤ C ′ Z R − R e / (ln( r ) − ln( R )) dr ≤ C / R − ( Z R − R e dr ) / ( Z R − R (ln( r ) − ln( R )) dr ) / ≤ C ′′ E R R − Case 3: s ≤ R/ 2. In this case there are two extra boundary terms in Green’sformula Z I R ( s es + t + (ln( R ) − 12 ln( s + t )) ∂ s e )However, e (0 , t ) lies on the boundary and is contained in a disk of radius R/ e (0 , t ) ≤ C ′′ E R for C ′′ sufficiently large and independent of R and s . We have the bound Z I R s s + t dt ≤ Z R 11 + t dt < ∞ which takes care of this term. To bound the second term we must bound Z I R ( ln ( R ) − 12 ln( s + t )) dt as in Case 2. Rescaling ( s, t ) to ( s/R, t/R ) and note that by continuity and the fact0 ≤ s ≤ / Z √ − s − √ − s ln( s + t ) dt < C ′′ for C ′′ is large. This implies that Z I R ( ln ( R ) − 12 ln( s + t )) dt ≤ C ′′ R as desired. Case 4: s ≥ R/ 2. In this case the whole disk of radius R/ H . We may apply Case 1 to conclude that e ( p ) ≤ C ′′ R − E R , after adjusting C ′′ . hus, we have demonstrated that if e ( z ) ≤ CR − on D R , we have e ( p ) ≤ C ′′ R − E R .Once we replace R by R/ e ( z ) ≤ CR − on D R we have e ( p ) ≤ C ′′ R − E R as desired.5.8. Completing the Proof - A Continuity Argument. We are now in positionto remove the pointwise assumption on e and replace it by an integral energy assump-tion. This will allow us to complete the proof of theorem 8. For convenience, let usabstract the relevant setup. For a given R > 0, let e : D R ( p ) ⊂ H + → [0 , ∞ )be a continuous function and let E D R ( p ) = Z D R ( p ) e for all D R ( p ) ⊂ D R ( p ). Suppose, e ( p ) ≤ C ′′ E D R ( p ) /R whenever e ≤ C/R on D R ( p ). Theorem 9. Let ~ = C/ C ′′ . For any D R ( p ) ⊂ D R ( p ) with E D R ( p ) ≤ ~ , we have e ( p ) ≤ CC ′′ E D R ( p ) /R Proof. The proof of this theorem is variant of a continuity argument which we repro-duce for completeness. For notational simplicity we will shift H + so that p = 0 andrescale e so that C = 1. On D R (0), let ρ ( r ) = ( R − r ) sup z ∈ D r (0) e ( z )for r ∈ [0 , R ]. Since ρ is a continuous function on a compact domain it must havea maximum at some r > 0. Let z ∈ D r be a point where ρ ( r ) = ( R − r ) e ( z ).Note that r = | z | . If ρ ( r ) ≤ / 4, then on D R/ (0),sup z ∈ D R/ e ( z ) ≤ /R and from our hypothesis we can deduce that e (0) ≤ C ′′ E D R/ (0) / ( R/ ≤ C ′′ E D R (0) /R as desired. Thus, we may assume ρ ( r ) = ( R − | z | ) e ( z ) > / et s = e ( z ) . We have s < ( R − | z | ) / D s ( z ) ⊂ D R (0).Since ρ ( | y | ) ≤ ρ ( | z | ) for any y ∈ D s (0), we have e ( y ) ≤ e ( z )( R − | z | ) ( R − | y | ) ≤ e ( z )( R − | z | ) ( R − | z | − R/ | z | / = 4 e ( z ) = 14 s Therefore, for y ∈ D s ( z ) the hypothesis e ( y ) ≤ s is valid and implies e ( z ) ≤ C ′′ E D s ( z ) /s = 16 C ′′ e ( z ) E D s ( z ) ≤ C ′′ e ( z ) E D R (0) < e ( z )since by hypothesis E D R (0) ≤ / C ′′ . This is a contradiction and thus ρ ( z ) ≤ / (cid:3) Note that ~ above is independent of R . Therefore, we have completed the proofof theorem 8. 6. Regularity and Convergence Interior Regularity for the ASD equation. The goal of the present sectionis to establish regularity results for the matched equations. We will begin with areview of the proof of interior regularity for the ASD equation. This material israther standard and covered in many sources (see [3]). We have chosen to include abrief discussion to facilitate the treatment of the rather involved regularity result forthe matched equations. Let X be a smooth Riemannian 4-manifold and let A besome fixed smooth SO (3)-connection. Let A be an ASD connection: F + A = 0As a stationary point of the Yang-Mills functional, A is automatically a Yang-Millsconnection d ∗ A F A = 0Recall that A is in Coulomb gauge with respect to A if d ∗ A ( A − A ) = 0Assume that A is ASD and in Coulomb gauge with respect to some fixed smooth A .We will tailor the dicussion to the case when X = D R × Σ although the results havedirect generalization to any X . Since in this section we are dealing with the interiorcase, we assume that D R does not intersect the boundary. ix some p > 2. Let p = 1 /p , q = 2 p − / p = 2 p − / 2. Note theembedding L /q ⊂ L /p Here is the main result we will need: Lemma 26. If A is L /p , then A is in L /q . If A ∈ L pk for k > , then A ∈ L pk +1 .Take any R ′ < R . Suppose we are given a sequence of ASD connections A i on X inCoulomb gauge with respect to some fixed A . If A i converges in L /p on D R × Σ then A i converges in L /q on any D R ′ × Σ . For any k > , suppose A i converges in L pk on D R × Σ . Then, A i converges in L pk +1 on any D R ′ × Σ . Using this lemma, one may immediately deduce regularity and convergence prop-erties of a sequences of ASD connections on X . We will discuss this in more detail atthe end of the section. The proof of this lemma occupies the rest of this subsection.Let A = A + B be an ASD connection in Coulomb gauge with respect to A . Thus(59) d ∗ A B = 0(60) F A + B = F A + d A B + B ∧ B If we project to the self-dual part of the curvature, we obtain(61) ( d + A + d ∗ A ) B = − F A − ( B ∧ B ) + Now, applying(62) 2 d ∗ A + d A : Ω + ( X ; g ) ⊕ Ω ( X ; g ) → Ω ( X ; g )we obtain that(63) ∆ B = B ′ · B + g Where ∆ = d ∗ A d A + d A d ∗ A is the Hodge Laplacian, g is a smooth function thatdepends only on A , B B ′ is the action of a smooth first order operator acting on B and B ′ · B is some algebraicmultiplication. Since we will be concerned with estimates on D R ′ × Σ, let ρ : D R → R be a bump function with support in D R such that ρ = 1 on D R ′ . We have(64) ∆ ( ρB ) = B ′ · ( ρB ) + g + L ( B )where L is some first order differential operator that depends on ρ . In the case when k = 1, this equation must be interpreted in the weak sense. In other words, givenany smooth section s of Ω ( X ; g ) with support in D R × Σ, we have h ρB, ∆ s i = h B ′ · ( ρB ) + g + L ( ρB ) , s i irst, we obtain L /q -regularity for ρB . For this, we use the embedding in equation(14) to obtain an L /q bound on B ′ · ( ρB ). Now, we apply the regularity results oftheorem 7 to obtain L /q -bounds on ρB .Now, we assume that we are in the stable range pk > k > 1. The embed-ding L pk · L pk − → L pk − implies an L pk − bound on B ′ · ρB and hence elliptic estimates give an L pk +1 -bound on ρB .The sequential version of this argument follows a similar pattern. First, B i − B j satisfies ∆ ( ρ ( B i − B j )) = B ′ i · ( ρB i ) − B ′ j · ( ρB j ) + L ( B i ) − L ( B j )= B ′ i · ( ρ ( B i − B j )) − ( B ′ j − B ′ i ) · ( ρB j ) + L ( B i ) − L ( B j )(65)Now, arguing as above using the Sobolev embeddings, we may conclude that A i converges on D R ′ × Σ as desired.6.2. Regularity For J-Curves in a Banach Space. We now turn to the discussionof regularity for holomorphic curves with values in a Banach space. Let B p = L p (Σ) × R N and let B p = B p ⊕ B p . We will assume that B p has a smooth almost complex structure J : B p → End ( B p , B p )Furthermore, we assume that, along L = 0 ⊕ B p , J is given by J where J ( b , b ) = ( − b , b )Thus, L is totally real with respect to J .In this section, take D R the be centered at the origin and let p > ≥ u : D R → B p such that u ∈ L pk ( D R ; B p ). Furthermore, assume that u | ∂D R maps to L . Such a map u is said to be J -holomorphic if ∂ J ( u ) = ∂ t u + J ( u ) ∂ s u = 0Here is the basic technical result we will need: emma 27. Let k > . Given u ∈ L pk ( D R ; B p ) with ∂ J u = g ∈ L pk ( D R ; B p ) we have u ∈ L pk +1 ( D R ; B p ) . Given a sequence u i ∈ L pk converging on L pk ( D R ) we have that u i converges in L pk +1 ( D R ′ ; B p ) for any R ′ ⊂ R .Proof. Our strategy is to reduce the problem to a regularity result for the Laplacian.Write u = ( u , u ) using the decomposition B p = B p ⊕ B p . By assumption, u vanisheson ∂D R . On the other hand, u satisfies the normal boundary condition ∂ t u = g | ∂D R .Applying ∂ t − J ( u ) ∂ s to ∂ J u = g we obtain ( ∂ s + ∂ t ) u = − ∆ u = − J ( u ) g ′ + g ′ + ( J ′ ( u )( u ′ )) · u ′ Here, as well as in the sequel, we will use u ′ to denote some first order differentialoperator on u with smooth coefficients. By assumption, J ( u ) g ′ ∈ L pk − and( J ′ ( u )( u ′ )) · u ′ ∈ L pk − in view of the product theorem L pl · L pl → L pl as long as p > l ≥ 1. Elliptic estimates from section 3 . 3, imply that u ∈ L pk +1 ( D R ; B p )as desired. Consider now the sequential version. For any given point p ∈ D R Wewill produce a uniform bound on a neighborhood D r ( p ) of p . By assumption, u i ( p )converge in B p . We may therefore, take r sufficiently small that J is uniformlybounded in C k +3 on the image of each ( u i ) | D r ( p ) . As above, we have∆ u i = − J ( u i ) g ′ i + g ′ i + ( J ′ ( u i )( u ′ i )) · u ′ i In view of the uniform bound on J ( u i ) in C k +3 , we have uniform bounds on the L pk − -norm of ( J ′ ( u i )( u ′ i )) · u ′ i . We now apply the regularity estimates to obtain a uniformbound on D R ′ . (cid:3) We now address the case when k = 1. We have: Lemma 28. Take p > . Given u ∈ L p ( D R ; B p ) with ∂ J u = g ∈ L p ( D R ; B p ) then u ∈ L p/ ( D R ; B p/ ) . Given a sequence u i ∈ L p converging on L p ( D R ) , then u i converges on L p/ ( D R ′ ; B p/ ) .Proof. We imitate the proof of the result above. However, this time the producttheorem maps L p ( D R ; B p ) × L p ( D R ; B p ) → L p/ ( D R ; B p ) → L p/ ( D R ; B p/ )The last embedding is necessary since we have not developed regularity theory formixed spaces such as L p ( D R ; B q ) where p = q . (cid:3) emark. The lemma above will be most useful to us when p > 4. To address thecase when 4 ≥ p > Lemma 29. Fix some p ′ > p . Given u ∈ L p ( D R ; B p ) ∩ L p ′ ( D R ; B p ′ ) with ∂ J u = g ∈ L p ′ ( D R ; B p ′ ) then u ∈ L p ′ ( D R ; B p ′ ) . Given a sequence u i ∈ L p ( D R ; B p ) ∩ L p ′ ( D R ; B p ′ ) converging in L p ( D R ; B p ) ∩ L p ′ ( D R ; B p ′ ) then u i converges in L p ′ ( D R ′ ; B p ′ ) .Proof. We will prove regularity around an arbitrary point x in ∂ ( D R ). For conve-nience, we take x = 0. Let T ( s, t ) = J − J ( u ( s, t )). By construction T (0 , 0) = 0. Let ρ be a bump function with support in D R and ρ = 1 on some D R ′ . Let v = ρu . Wehave that ∂ J ( ρu ) ∈ L p ′ with support in D R . Since v has support away from S R of D R , we may view v as afunction on the closed disk D R with Lagrangian boundary conditions. For this oneneeds to ”round” the corners of D R but since the support of v vanishes around thereit does not affect the argument. On D R , v satisfies ∂ J v + T ( s, t ) ∂ t v = 0Assuming that the support of ρ is sufficiently small, the norm of T : L p ′ ( D R ; B p ′ ) → L p ′ ( D R ; B p ′ )as well as T : L p ( D R ; B p ) → L p ( D R ; B p )is small. It follows that the operator ∂ J + T ( s, t ) ∂ t is surjective as an operator L p → L p as well as L p ′ → L p ′ and the kernel consists of constant solutions on theLagrangian L . This implies that v ∈ L p ′ as desired. The convergence argument issimilar. (cid:3) Regularity for the Matched Equations. We now turn to the regularity re-sults for the case of matched boundary conditions. Let D R ⊂ C be a disk centeredat the origin. For some p > 2, consider a J -curve u : D − R → M with u ∈ L p ( D − R ). and an ASD connection A ∈ L p ( D R × Σ) on D R × Σ. We decompose A as A = α + φds + ψdt with α ( s, t ) ∈ A . We will assume that ( u, A ) are matched at the boundary u ( s, 0) = [ α ( s, α ( s, t )] denotes the equivalence class in M . By the Coloumb slice theorem (see[4]), there exists a smooth connection A on D R × Σ and a gauge transformation g ∈ L p with the following properties. If B = A − A , then, d ∗ A ( B ) = 0 nd B ( s, ∂ t ) = 0Here is our main regularity theorem: Theorem 10. Let ( u, A ) be a matched pair on D R × Σ . Assume u ∈ L p and A ∈ L p for p > . Furthermore, assume that u is J -holomorphic and A is ASD. If A is inCoulomb gauge with respect to a smooth connection A , then ( u, A ) is smooth. The proof this result occupies the rest of this section. Our strategy is to proveregularity for the different components of A separately. A variant of this strategywith for a different boundary value problem appears in [5].6.3.1. Estimates on ψ . Let us decompose A as A = α + ψ + φ Since ∆ B + B · B ′ = 0, we may project to the dt -component to deduce that∆ ( ψ − ψ ) + L ( B · B ′ ) = 0where L ( B · B ′ ) is the projection of B · B ′ to the dt -component. By assumption, ψ (0 , s ) = ψ (0 , s ). Since ψ is smooth we are in position to apply the Dirichletboundary value problem estimates to deduce regularity of ψ . We summarize thebootstrapping estimates with the lemma below: Lemma 30. Let p = 1 /p , q = 2 p − / and p = 2 p − / . If A ∈ L p ( D R × Σ) ,then || ψ || L /p ( D R ′ × Σ) ≤ C p, || A || L /p ( D R × Σ) and || ψ || L /q ( D R ′ × Σ) ≤ C p, || A || L /p ( D R × Σ) If A ∈ L pk ( D R × Σ) for k > , then || ψ || L pk +1 ( D R ′ × Σ) ≤ C p,k || A || L pk ( D R × Σ) Furthermore, the constants C p,k do not depend on the choice of A .Proof. The proof of this proposition is identical to that of the interior case aside fromthe fact that now we base our linear elliptic estimates on the Dirichlet problem whichwe discussed in section 3 . 1. The nonlinear estimates on B ′ · B is identical to the onefor the interior case and gives rise to the same estimates. (cid:3) .3.2. Estimates on φ . We now address the regularity of φ . This case is a bit moresubtle in view of the boundary condition. Since A is ASD, we have ∗ F α + ∂ s ψ + ∂ t φ + [ ψ, φ ] = 0By assumption, ψ is smooth at the boundary. In addition, since ( u, A ) are matched, ∗ F α vanishes at the boundary. The basic strategy now is to apply the elliptic theoryfor the Neumann problem to obtain regularity for φ . However, we must be carefulsince we are initially starting with an L p -configuration and we must discuss a weakversion of the Neumann boundary value problem. Let us first introduce some nota-tion. Let I τ the set of points ( s, τ ) ∈ D R . Thus, I × Σ is the matched boundary of D R × Σ. Let f be a smooth function with support in D R × Σ such that ∂ t f = 0 on I × Σ. We now check that φ satisfies a weak version of the Neumann problem: Lemma 31. h φ, ∆ f i D R × Σ = h ∆ φ, f i D R × Σ + h [ φ, ψ ] , f i I × Σ where ∆ φ is the linear projection onto the ds -component of B · B ′ .Proof. Let D r { ( s, t ) ∈ D R | t ≥ r } Since φ is smooth on D r for r > 0, we obtain h φ, ∆ f i D r × Σ = h ∆ φ, f i D r × Σ + h φ, ∂ t f i I τ × Σ + h∗ F α , f i I τ × Σ + h [ φ, ψ ] , f i I τ × Σ (cid:3) We need to argue that the last 3 terms approach 0 as τ → 0. For h φ, ∂ t f i I τ × Σ + h [ φ, ψ ] , f i I τ × Σ this is straightforward since ∂ t f = 0 on I × Σ and φ = 0 on I × Σ.We need to examine the term h∗ F α , f i I τ × Σ . Since A ∈ L p , we have φ ∈ L p ( D ; L p (Σ)) ⊂ C ( D ; L p (Σ))The moment map sending α to F α is continuous as a map L p (Σ) → L p − (Σ). Therefore, ∗ F α ∈ C ( D, L p − (Σ))By assumption, F α = 0 on I × Σ.Since f is smooth, it gives a well defined element of C ( D ; L p (Σ)). And h∗ F α , f i I τ × Σ ≤ C · max I τ | F α | L p − where C depends only of f . Therefore, h∗ F α , f i I τ × Σ → τ → emma 32. Let p = 1 /p , q = 2 p − / and p = 2 p − / . If A ∈ L p ( D R × Σ) ,then || φ || L /p ( D R ′ × Σ) ≤ C p, || A || L /p ( D R × Σ) and || φ || L /q ( D R ′ × Σ) ≤ C p, || A || L /p ( D R × Σ) If A ∈ L pk ( D R × Σ) for k > , then || φ || L pk +1 ( D R ′ × Σ) ≤ C p,k || A || L pk ( D R × Σ) Furthermore, the constants C p,k do not depend on the choice of A .Proof. This time our estimates are based on the solution to the Neumann problemthat we described in section 3 . 2. Given φ , we assume that φ satisfies h φ, ∆ f i D × Σ = h g, f i D × Σ + h ρ, f i I × Σ for all smooth f with ∂ t f = 0 on I × Σ and support in D × Σ. One obtains estimateson φ from the regularity on g and ρ . On our situation, ρ = [ ψ, φ ] I × Σ and g = B ′ · B .In view of the regularity results we obtained on ψ , the estimates on φ from combiningmore details. (cid:3) Slicewise Estimates on α . The starting observation is that the ASD equationtogether with the Coulomb gauge condition imply that( d + d ∗ )( α − α ) = B · B + φ ′ + ψ ′ + g where g is some fixed smooth function. In other words, ( d + d ∗ )( α − α ) does notinvolve any ( s, t )-derivatives of α . We have the following estimates: Lemma 33. If A ∈ L p ( D R × Σ) then || α || L /p (Σ; L /p ( D R )) ≤ C p, || A || L p ( D R × Σ) and || α || L /q (Σ; L /q ( D R )) ≤ C p, || A || L p ( D R × Σ) If A ∈ L pk ( D R × Σ) for k > then || α || L pk +1 (Σ; L p ( D R ′ )) ≤ C p,k || A || L pk ( D R × Σ) Furthermore, the constants do not depend on A .Proof. The proof of this proposition combines slicewise elliptic regularity with esti-mates on the nonlinear terms. To begin, if B ∈ L p , then by equations (15) and(17) B · B ∈ L /p ( D R × Σ) ∩ L /q ( D R × Σ)We have already established in lemma 30 and 32 that ψ, φ ∈ L /p ( D R × Σ) ∩ L /q ( D R × Σ) hus, we have( d + d ∗ )( α − α ) ∈ L /p ( D R × Σ) ∩ L /q ( D R × Σ) ⊂ L /p (Σ; L /p ( D R )) ∩ L /q (Σ; L /q ( D R ))We now apply elliptic regularity for d + d ∗ with values in the Banach space L /p ( D )(or L /q ( D R )) to deduce that α − α ∈ L /p (Σ; L /p ( D R )) ∩ L /q (Σ; L /q ( D + ))To obtain the higher estimates, one proceeds in a similar fashion. We have B · B ∈ L pk ( D R × Σ) and φ, ψ ∈ L pk +1 ( D R × Σ). Therefore,( d + d ∗ )( α − α ) ∈ L pk ( D R × Σ) ⊂ L pk (Σ; L p ( D R ))and by lemma 27 we get α ∈ L pk +1 (Σ; D R ). Finally, the uniform bounds on ψ , φ aswell as A on D R ′ × Σ yield inform bounds on α on D R ′ × Σ. (cid:3) t, s ) -Estimates on u , α . Consider now the equations(67) ∂u = 0(68) ∂ t α + ∗ ∂ s α = d α φ + ∗ d α ψ It will be convenient at this point to treat the pair ( u, α ) as a map from D R . For this,define v : D R → M as v ( s, t ) = u ( s, − t ). We may now view the pair ( v, α ) as a map D R → M − × A with the Lagrangian boundary condition ( v (0 , s ) , α (0 , s )) ∈ L . The estimates belowwill follow by applying our regularity results for Banach valued holomorphic curves. Lemma 34. Assume ( u, A ) ∈ L pk with p > and k > . We have ( u, A ) ∈ L pk +1 .Assume ( u, A ) ∈ L p with p > . We have ( u, A ) ∈ L p/ .Assume ( u, A ) ∈ L p with p > . We have ( u, A ) ∈ L /p .Proof. To illustrate the proof let us prove the second claim. The proofs of the otherparts are similar. By lemma 30 and 32, we have ψ, φ ∈ L /p . We have d α φ + ∗ d α ψ ∈ L /p Applying the change of coordinates from section 4 . β = ( v ′ , α ′ ) : D R → B p such that ∂ t β + J ( β ) ∂ s β = γ where γ is d α φ + ∗ d α ψ in the new coordinates. Since the change of coordinatespreserves the L /p -structure, we may use the elliptic regularity lemma 28 to deduce hat β ∈ L /q . Applying the change of coordinates in the other direction, we obtain( u, α ) ∈ L /q ( D R ; L /q (Σ)) as desired. (cid:3) Synthesis of Regularity Arguments. We are now in position to put togetherthe estimates from the previous parts to obtain regularity for the matched equations.Assume we have a matched pair ( u, A ) on D R × Σ. We will obtain regularity andbounds on various Sobolev norms in a small neighborhood of the center of D R . Notethat the size of the neighborhood will depend on the specific norm in question. Infact, at each elliptic estimate we need to shrink the size of R . Since we are interestedin a regularity/compactness statement on a given compact set, it suffices to provethat each point p ∈ D R has a neighborhood where any given L pk -norm is bounded. Step 1 : Assume ( u, A ) ∈ L p = L /p where p = 1 /p < / 2. We claim that( u, A ) ∈ L / (2 p − / as long as p > / 4. Let p = 2 p − / q = 2 p − / ψ ∈ L /q and ψ ∈ L /p . Now, we use lemma32 to deduce that φ ∈ L /q and φ ∈ L /p . Next, we use lemma 33 to deduce that α ∈ L /p (Σ; L /p ( D R )). Finally, we use lemma 34 to deduce that α ∈ L /p ( D R ; L /p (Σ))and u ∈ L /p . We obtain, using section 2 . α ∈ L /p ( D R × Σ) as desired. Let δ = p − p = 1 / − p > p k = 2 p k − − / δ k = 1 / − p k − > δ Thus, after finitely many steps, 1 / < p k < / Step 2 : Assume ( u, A ) ∈ L /p where 1 / < p = 1 /p < / 4. Let q = 2 p − / 4. Weclaim that ( u, A ) ∈ L p/ . First, we apply lemma 30 to deduce that ψ ∈ L /q .Now, we use lemma 32 to deduce that φ ∈ L /q . Next, we use lemma 33 todeduce that α ∈ L /q (Σ; L /q ( D R )). Finally, we use lemma 34 to deduce that α ∈ L p/ ( D R ; L p/ (Σ)) and u ∈ L p/ . We have ( u, A ) ∈ L p/ as desired. Step 3 : Assume ( u, A ) ∈ L pk where pk > p > k > 1. We claim that( u, A ) ∈ L pk +1 . First, we apply lemma 30 to deduce that ψ ∈ L pk +1 . Now, weuse lemma 32 to deduce that φ ∈ L pk +1 . Next, we use lemma 33 to deduce that α ∈ L pk +1 (Σ; L p ( D R )). Finally, we use lemma 34 to deduce that α ∈ L pk +1 ( D R ; L p (Σ)) nd u ∈ L pk +1 . We have ( u, A ) ∈ L pk +1 as desired.This completes the proof of theorem 10. Remark. Given a sequence ( u i , A i ) converging uniformly in L p we may conclude that,for some subsequence, ( v (0 , , α (0 , L p to a limit x ∈ M × A .We may therefore choose a universal chart for all sufficiently large i where we canstraighten the Lagrangian boundary conditions using section 4 . 2. Finally, we useelliptic estimates from section 6 . i sufficiently large. This provides a sequential version of theorem 10.7. Compactness Review of Weak Compactness. Our proof compactness will use the funda-mental results of Uhlenbeck (see [18] as well as the refinements in [4]). Let X be anoriented, compact, Riemannian 4-manifold possibly with boundary. Let A i be a L p sequence of connections on some principal bundle (or associated vector bundle) witha compact structure group G . Theorem 11. Take p > , and assume that || F A i || L p is bounded. Then, there existsan L p connection A , a subsequence A j ⊂ A i and L p gauge transformations g j withthe following properties:1. g i ( A i ) are in some fixed L p ( X ) neighborhood of A and converge L p -weakly to A g i ( A i ) converge strongly to A in L q for any < q < p − p The compactness theorem is useful in conjunction with the following gauge fixingresult: Theorem 12. Let A i be a sequence of L p ( X ) connections converging to A in the weak L p topology. Then, there exist L p ( X ) gauge transformations g i , such that g i ( A i ) arein Coloumb-Neumann gauge with respect to A . In other words, d ∗ A ( g i ( A i ) − A ) = 0 ∗ ( g i ( A i ) − A ) | ∂X = 07.2. Interior Compactness. Let us briefly recall the compactness results of Gro-mov and Uhlenbeck. These are well known and discussed in detail in many texts (seefor instance [14] and [3]). It is worth mentioning that the a priori estimates of thiswork give an independent proof of these compactness results.Let us first discuss the case of Uhlenbeck compactness. Fix X as above, and let ◦ = X − ∂X . Let A i be a sequence of ASD connections on X ◦ . By the regular-ity results, we may assume that, after a gauge transformation, the A i are smooth.Suppose that A i have uniformly bounded energy. Theorem 13. There exists a finite sequence of points p k ∈ X ◦ and a subsequence A j with the following properties:1. A j converge (in any C k -norm) on compact subsets of X ◦ − ∪ k p k to an ASDconnection A ∞ .2. If ∪ k p k is nonempty, there exists ~ > , independent of A i such that E ( A ∞ ) < lim inf E ( A j ) − ~ Now, we turn the Gromov compactness. Let D be any compact Riemann surface(possibly with boundary). Let ( M, ω, J ) be a compact symplectic manifold withcompatible almost complex structure J . Consider a sequence of J -holomorphic maps u i : D ◦ → M with uniformly bounded energy. As in the ASD case, such maps are automaticallysmooth as soon as they are L p for p > 2. Here is the version of Gromov compactnesswe need: Theorem 14. There exists a finite sequence of points p k ∈ D ◦ and a subsequence u j with the following properties:1. u j converge (in any C k -norm) on compact subsets of D ◦ − ∪ k p k to a holomor-phic curve u ∞ .2. If ∪ k p k is nonempty, there exists ~ > , independent of u i such that E ( u ∞ ) < lim inf E ( u j ) − ~ Definition 18. A singular set S on D R × Σ is a finite collection of points x i ∈ ( D − R − ∂D − R ) , y i ∈ ( D R − ∂D R ) × Σ , z i × Σ ∈ ( I R − ∂I R ) × Σ . The z i × Σ are the boundary slices of S . Theorem 15. Assume that we have a uniform bound E ( u i , A i ) < C . There exists asubsequence ( u j , A j ) and a singular set S with the following properties. Let K be acompact set in ˚ D − R − S and K be a compact set in ˚ D R × Σ − S . We have that u j converges in any C k norm on K and A j converges in any C k -norm on K . Finally,the energy loss at each singular point is at least ~ for some sufficiently small ~ > independent of the choice of sequence.Proof. Our first task is to argue that outside some finite singular set we have E ( D R ) ≤ ~ for all sufficiently small R < R . Let us call p ∈ D R singular if for any D r with enter p we have lim inf E D r ( u i , A i ) ≥ ~ Suppose p is such a point. Pass to a subsequence ( u j , A j ) wherelim E D r ( p ) ( u j , A j ) ≥ ~ for any r > 0. Now, consider a different singular point for ( u j , A j ). Let us call it p . We may pass to a subsequence ( u k , A k ) such that lim E D r ( p ) ( u j , A j ) ≥ ~ for any r > 0. Repeating this N times yeilds N singular points as well as a subsequence whichhas at least ~ energy near each singular point. Since E ( u i , A i ) is bounded, there canbe at most a finite number of such singular points. Thus, we may restrict to provingcompactness away from these singular points. We therefore, consider a disk D R × Σwhere E ( u i , A i ) ≤ ~ . Now, we may apply the a priori estimates of section 5 to obtainan L ∞ bound on du i as well as bounds on || F A i || L , ||∇ A i F A i || L . The results ofUhlenbeck in section 7.1 imply that we can put A i in a Coloumb-Neumann gaugewith respect to some smooth connection A ∞ . The lemma above implies convergenceof A i in any L p with p < 4. Since in dimension 2 the map L p → C is compact forany p > 2, we obtain a C convergent subsequence for u i . We are now in position toapply the regularity and convergence results of section 6.3 to conclude that we haveuniform C k -bounds on ( u i , A i ) in a neighborhood of each point. This implies thatafter passing to a subsequence we have C k − -convergence of ( u i , A i ) on any compactset. (cid:3) Gromov-Uhlenbeck Compactness. We can now state and prove our gener-alization of the results above. We will consider a sequence of matched pairs ( u i , A i )on D R × Σ. By our regularity results, we may assume that the sequence consistsof smooth elements. The following convergence result is useful in our discussion ofGromov-Uhlenbeck compactness. Lemma 35. Suppose A i are in Coloumb-Neumann gauge with respect to some fixedsmooth A on D R × Σ . Furthermore, assume that we have a uniform bound on ||∇ A i F A i || L and || A i − A || L . We have that A i has a strongly L p -convergent subse-quence for any ≤ p < on D R/ × Σ .Proof. By Kato’s inequality (see equation (57)), the bound on ||∇ A i F A i || L gives auniform bound on || F A i || L . The embedding L · L → L implies that the L -bound on A i gives us an L -bound on A i ∧ A i . Now, ∇ F A i = ∇ A i F A i + ( A − A i ) · F A i In view of the L -bound on F A i , we obtain an L -bound on ∇ F A i . Since d ( A i − A ) = F A i − ( A i − A ) ∧ ( A i − A ) − F A e obtain a uniform bound on || d A i || L . The embedding L → L p is compact for all p < 4, therefore d A i is strongly precompact in L p . Now, since A i are in Coloumb-Neumann gauge, we use the estimate || A i − A j || L p ≤ C ( || A i − A j || L p + || d ∗ ( A i − A j ) || L p + || d ( A i − A j ) || L p )on D R/ × Σ to deduce that A i is strongly precompact in L p ( D R/ × Σ). (cid:3) Removal Of Singularities Statement of results. In previous sections we have discussed a compactnesstheorem in the context of matched pairs ( u, A ). As demonstrated, a sequence of pairswith a uniform energy bound converges outside a set of singularities. Thus, such asequence gives rise to a matched pair ( u ∞ , A ∞ ) that has finite energy but is not de-fined on the entire domain. For interior singular points x i and y i of definition 18, wecan complete the pair ( u ∞ , A ∞ ) using removal of singularities for J -curves and ASDequations (see [15] and [3]). It remains to address the singularities at the boundaryslices z i .Let D R be the disk D R = { ( s, t ) ∈ C | s + t ≤ R , s ≥ } and let D ∗ R = D R − (0 , u, A ) defined on D ∗ R as in section5 . 2. We assume E ( u, A ) < ∞ . Theorem 16. There exists a matched pair ( u ′ , A ′ ) on D R that is gauge equivalent to ( u, A ) on D ∗ R . The pair ( u ′ , A ′ ) is said to extend ( u, A ) . This theorem completes our framework of Gromov-Uhlenbeck compactness. Wesee that a sequence of pairs weakly converges to a limiting pair. In case some ~ > The Chern-Simons Functional. Given a closed 3-manifold Y with a U (2)-bundle P and connection (on the associated SO(3)-bundle) A we may define theChern-Simons invariant as follows. Pick a flat base connection A and let B = A − A .Chern-Simons invariant is defined as(69) CS ( A ) = tr ( Z Y B ∧ d A B + 23 B ) = tr ( Z Y B ∧ F A − B ) he importance of Chern-Simons for us comes from the following. Let Y = ∂X andassume A extends to a flat connection on X and A extends to a connection on X .We have CS ( B ) = tr ( Z X d A A ∧ d A A + 23 (( d A A ) A − A ( d A A ) A + A d A A ))= tr ( Z X d A A ∧ d A A + 2 Ad A A )= tr ( Z X F A ∧ F A )(70)Now, assume that Y = Σ × S and take some flat connection α on Σ. We can pullit back to obtain A on Y . If B = α + φds then, B = α + [ α, φ ] dsB = α [ α, φ ] ds + φα dsd A B = d α α + d α φds − ∂ s αdsd A B ∧ B = ( d α α ) φds + ∂ s α ∧ αds − ( d α φ ) αds Since tr ( B ) = 3 tr ( φα ) ds and d α ( φα ) = ( d α φ ) αds = φd α αds and we get(71) CS ( α + φds ) = Z S Z Σ tr ( φF α ) + tr ( ∂ s α ∧ α )Thus, if F α = 0 or φ = 0, we are reduced to the canonical 1-form of symplecticgeometry.Consider now a matched pair ( u, A ). By taking R small, we may assume E ( u, A )is as small as we wish. We regard A as living on D + R × Σ where D + R is the positivehalf disk and u as defined on D − R . Take ( r, φ ) to be polar coordinates on C . Given( r, φ ) ∈ D ∗ R/ we take D r ( r, φ ) centered at ( r, φ ) that is completely contained in D ∗ R .If the energy for D R is small, we apply theorem 8 to obtain || F A ( r, φ ) || L (Σ) ≤ Cr − ( E ( u, A )) / and | du ( r, φ ) | ≤ Cr − ( E ( u, A )) / Thus, given ǫ > r sufficiently small, || F A ( r, φ ) || L (Σ) ≤ ǫ/r and | du ( r, φ ) | ≤ ǫ/r .It follows that on S − r , u is contained in a single chart where it can be written as α + α with α a fixed flat connection and α some 1-form. We choose a chart where α ( r, π/ 2) = 0 and d ∗ α α = 0. With this choice of lift we have | du | ≥ C ′ |∇ α α | e may trivialize A to have the form α + α + + βdr with no dφ component. By thematching condition, we can assume α + (0 , π/ 2) = 0. Thus, the connections on thetwo sides coincide at that point. On the other hand, we have g ∗ r ( α + α − ( r, − π/ α + α + ( r, − π/ g r is a gauge transformation of connections on Σ. Energy of the connection isthen expressed as Z D + r || F A || + ρ − || ∂ φ A || ρdρdφ This implies that || ∂ φ A || ≤ ǫ . Thus, || A ( π/ , r ) − A ( − π/ , r ) || L ≤ ǫ for all r small. Similarly, the energy of u is exwe have || α − ( π/ , r ) − α − ( − π/ , r ) || L ≤ ǫ Lemma 36. Given g ∈ G (Σ) there exists an extension ˜ g ∈ G (Σ × [0 , such that ˜ g |{ } = g , ˜ g |{ } = Id and || ˜ g || L ≤ C || g || L for some universal C > .Proof. This result is based on a theorem of Hang and Lin [19] and is discussed indetail in [6]. (cid:3) Setting τ = ˜ g ∗ ( α − + α − ( ρ, − π/ , × Σ suchthat || τ − α − ( ρ, − π/ || L ;[0 , × Σ ≤ C ′ || α − ( ρ, − π/ − α + ( ρ, − π/ || L ;Σ By taking a close approximation (in L ) of ˜ g , we may assume that it is constand nearthe boundary in the transverse direction. This is helpful for patching connectionstogether.We built a closed 3-manifold Y r as follows. Join S + r × Σ with S − r × Σ along ( r, π/ × Σ.Glue in Z = [0 , × Σ by identifying 0 × Σ with ( r, − π/ × Σ on S − r and 1 × Σ with( r, − π/ × Σ on S + r . By construction we obtain a connection A ′ on the 3-manifold Y r . The Chern-Simons on Y r with respect to α is given by − Z [0 , × Σ tr ( τ − α ) + Z π/ − π/ Z Σ tr ( α + ∧ ∂ φ α + ) dφd Σ + Z π/ π/ Z Σ tr ( α − ∧ ∂ φ α − ) dφd ΣWe have | Z [0 , × Σ tr ( τ − α ) | ≤ ( Z π/ − π/ Z Σ | ∂ φ α + | ) + ( Z π/ π/ Z Σ | ∂ φ α − | ) since | du | controls any L p -norm of the lift α . Since α ± ( π/ , r ) = 0 we obtain Z π/ π/ Z Σ tr ( α − ∧ ∂ φ α − ) dφd Σ = Z π/ π/ Z Σ Z φπ/ tr ( ∂ v α − ∧ ∂ φ α − ) dvdφd Σ π/ − π/ Z Σ tr ( α + ∧ ∂ φ α + ) dφd Σ = − Z π/ − π/ Z Σ Z φ − π/ tr ( ∂ v α + ∧ ∂ φ α + ) dvdφd ΣThus, we can bound CS ( A ′ ) by Cr ( Z S + r || F A || + r − || ∂ φ A || + Z S − r | du | ) = Cr∂ r E ( r ) ≤ ǫ Isoperimetric Inequality. From the previous section we have concluded thatfor the specific choice of Chern-Simons we have the estimate(72) CS ( A ′ ) ≤ Cr∂ r E ( r )To obtain the Isomperimetric Inequality we must now relate CS ( A ′ ) to E ( ρ ). Let D ± δρ = { ( s, t ) ∈ D + ρ | s + t ≥ δ } Given 0 < δ < ρ we define a 4-manifold X δρ by taking the union of all Y t for t ∈ [ δ, ρ ].Thus, X δρ consists of 3 pieces. D ± δρ × Σ and [0 , × [ δ, ρ ] × Σ. We define a connection˜ A on X δρ as follows. On D + δρ × Σ take A in the gauge where A = α + + βdr . On D − δρ × Σ we take a lift α − of u as above such that α + ( r, π/ 2) = α − ( r, π/ g ∗ r α − = α + at φ = − π/ 2. We take any smooth extension ˜ g r of g r to [0 , × [ δ, ρ ] × Σas above with the condition that at r = ρ the extension agrees with the one for Y r .On [0 , × [ δ, ρ ] × Σ we set the connection to be ˜ g r α − and extend β arbitrarily. On X δ,ρ we have Z X δ,ρ F A = E ( A D + δ,ρ ) + E ( u D − δ,ρ )since ˜ A is flat on [0 , × [ δ, ρ ] × Σ. Relating the energy to Chern-Simons of theboundary, we obtain Z X δ,ρ F A = CS ( A ′ ρ ) − CS ( A ′′ δ )where A ′′ is the restriction of ˜ A to [0 , × δ, × Σ. A priori, CS ( A ′′ ρ ) may differ thethe definition of the previous section by a multiple of 4 π . However, we see that CS ( A ′′ δ ) = E X δ,ρ − CS ( A ′ ρ ) is arbitrarily small when R is small and thus is specifieduniquely. Thus, by taking the limit as δ → CS ( A ′′ δ ) → CS ( A ′ r ) = E ( r ). This gives the desired inequality 72 and thus E ( r ) ≤ C ′ r β where β > Completing the Proof. So far we have deduced an energy decay E ( r ) ≤ Cr β for a matched pair on a punctured disk. Let us now use this decay to completetheorem 16. First, let us focus on the connection A : Lemma 37. There exists C > such that for all r sufficiently small: a ) sup φ || F A ( r, φ ) || L (Σ) ≤ C ′ r β − b ) sup φ || F A ( r, φ ) || L ∞ (Σ) ≤ C ′ r β − cos( φ ) − roof. To show a), begin by taking r small, we may assume that E ( A ) ≤ ~ . Given( r, φ ) ∈ D ∗ r / we take D r ( r, φ ) that is completely contained in D ∗ r . We now applytheorem 8 to obtain || F A ( r, φ ) || L (Σ) ≤ Cr − ( E ( A | D ∗ r )) / ≤ C ′ r β − as desired. For b), we take a point ( x, y ) on Σ and fix ( r, φ ). The 4 dimensional ball B r cos( φ ) / centered at ( x, y, r, φ ) is contained in D ∗ r × Σ and by 4-dimensional analysisof the ASD equation on a ball (see [3]) we obtain || F A ( r, φ ) || L ∞ (Σ) ≤ Cr − cos( φ ) − E ( A | D ∗ r ) ≤ Cr β − cos( φ ) − (cid:3) We now cite the following result from [6]: Theorem 17. Let A satisfy a) and b) from the previous lemma. For some p > ,there exists a gauge transformation g ∈ L p ( D ∗ × Σ) such that g ∗ A extends to an L p connection on D × Σ . By continuity such an extension A ′ must satisfy the ASD equation on D × Σ. Wenow turn to extending u . By our energy decay, we have | du ( r, φ ) | ≤ Cr β − as in the lemma above. 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Simons Center for Geometry and Physics, Stony Brook University, Stony Brook,NY 11794 E-mail address : [email protected]@gmail.com