aa r X i v : . [ m a t h . S G ] S e p ON REGULARITY FOR J -HOLOMORPHIC MAPS MAX LIPYANSKIY
Abstract.
We provide a short proof that an L and J -holomorphic curve is in factsmooth. As an application, we deduce a removal of singularity theorem for curvesof finite energy. Statement of Results
Given a manifolds Σ. Let L pk (Σ) denote the Sobolev space of functions with allderivatives up to order k in L p . Such functions may be valued in some finite dimen-sional vector space. Let M be a compact smooth manifold with a smooth almostcomplex structure J . For convenience, we will fix an embedding i : M ⊂ R N andassume that J extends to smooth almost complex structure in a neighborhood of M .Let Σ be a Riemann surface with complex structure j . We consider a map (definedalmost everywhere) u : Σ → M to be L if the corresponding map i ◦ u : Σ → M isin L . Furthermore, such a map is said to be J -holomorphic if du ◦ j = J ◦ du almost everwhere. Theorem 1.
Suppose u : Σ → M is L and J -holomorphic. We have u ∈ C ∞ ( R N ) . Let D be the unit open disk in the plane and let D ∗ be the punctured disk. As acorollary, we deduce: Theorem 2.
Suppose u : D ∗ → M is J -holomorphic and has E ( u ) = 12 Z D ∗ | du | < ∞ We have that u extends smoothly to a J -holomorphic map on D . There is a vast literature on L -regularity for harmonic maps going back to Morrey.See [3] for references. Acknowledgement.
We wish to thank Tom Mrowka for suggesting the key lemmain this paper. We also would like to thank the Simons Center For Geometry andPhysics for their hospitality while this work was being completed. . Proofs
Our proof of regularity will be based on the following application of Stokes’ theorem.
Lemma 1.
Let f, g ∈ L (Σ) be functions with compact support. We have df ∧ dg ∈ L − (Σ) .Proof. Take test function h ∈ L (Σ). Note that in since df , dg ∈ L (Σ), a priori df ∧ dg ∈ L (Σ) while h L ∞ (Σ). Therefore, it is not clear how to define R Σ h ∧ df ∧ dg .Take a 3-manifold Y with ∂Y = Σ. There exists ˜ f , ˜ g, ˜ h ∈ L / ( Y ) that extend thegiven f, g, h from Σ to Y . By solving the Dirichlet problem, such extensions canbe taken to depend continuously on the given f, g, h . Since L / ( Y ) → L ( Y ) indimension 3, we have d ˜ h ∧ d ˜ f ∧ d ˜ g ∈ L ( Y ). Finally, Stokes theorem implies that Z Σ hdf ∧ dg = Z Y d ˜ h ∧ d ˜ f ∧ d ˜ g for smooth h so we may set the RHS as the definition of the integral on the LHS. (cid:3) Remark.
As a slight generalization, note that one may take the functions to bematrix valued. In addition, the L − -norm of df ∧ dg depends only on the L -normsof df , dg and not on f , g . Indeed, taking replacing f with f + const we may assume || f || L ≤ C || df || .We now turn to the proof of the theorem. As the theorem is local in Σ we willfocus on the case of D r - a disk of radius r in the plane. Lemma 2.
Given u : D → M as above. If u ∈ L ,loc ( D ) , then u is smooth.Proof. Let ( s, t ) be coordinates on D . We will establish that u ∈ L / . Since L / → L p for p >
2, higher regularity is standard (see [1]) and follows from Sobolevmultiplication theorems.Let ∆ = ∂ s + ∂ t . Since ∂ s u + J ( u ) ∂ t u = 0, we apply ∂ s − J ( u ) ∂ t to deduce that∆ u + ∂ s J ∂ t u − J ∂ t J ∂ t u = 0This equation holds in the weak sense. Now, using the fact that J ∂ t J = − ∂ t J J and
J ∂ t u = − ∂ s u we deduce that ∆ u + ∗ dJ ∧ du = 0where ∗ is the Hodge star operator on Σ.We rewrite the equation as ∆ u + T ( u ) = 0. With T ( u ) = ∗ ( dJ ∧ du ). T defines acontinuous operator L / → L − / in view of the embedding L / · L / · L → L n addition, T defines a continuous operator L → L − . This follows from the previouslemma. By rescaling the disk, we may assume that the L -norm of dJ is as smallas we like. Therefore, the norm of T is as small as we like on the relevant Sobolevspaces.Take a bump function φ with support on D / . φu satisfies ∆( φu )+ T ( φu ) = g ∈ L .Since T is small, the solution to the Dirichlet problem implies there exists a unique v ∈ L / ( D ) such that ∆( v ) + T ( v ) = g and v = 0 on ∂D . Viewed as an element of L , v satisfies the same equation and thus v = φu . Therefore, u ∈ L / as desired. (cid:3) We now prove the removable singularities theorem. This follows from theorem 1together with the following standard lemma:
Lemma 3.
Given a bounded smooth u : D ∗ → R n with E ( u ) < ∞ , we have that u ∈ L as a map from D .Proof. Since u ∈ L ∞ , we need to check that du is the weak derivative of u on D . Takea bump function φ : D → R with φ = 1 outside the 1 / φ ǫ ( s, t ) = φ ( s/ǫ, t/ǫ ). Set u ǫ = φ ǫ u and note that u ǫ is smooth. Take asmooth test function f . We have h u, df i D = lim ǫ → h u ǫ , df i D = lim ǫ → h du ǫ , f i D = lim ǫ → h ( du ) φ ǫ + udφ ǫ , f i D Since lim ǫ → h ( du ) φ ǫ , f i D = h ( du ) , f i D , we need only show thatlim ǫ → h udφ ǫ , f i D = 0So see this, note that dφ ǫ has support on D ǫ and is bounded by Cǫ − for some uniform C >
0. Since u and f are bounded, h udφ ǫ , f i D ≤ C ′ πǫ ǫ − . (cid:3) References [1]
D McDuff , D Salamon J -holomorphic curves and symplectic topology Amer. Mathemat-ical Society, (1994).[2]
M Taylor
Partial differential equations I. Basic theory. , Applied Mathematical Sciences,115. Springer, New York, 2011.[3]
F Lin and C Wang
The Analysis of Harmonic Maps and Their Heat Flows. , WorldScientific, 2008.
Simons Center for Geometry and Physics
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