Schwarz Reflection Principle, Boundary Regularity and Compactness for J-Complex Curves

We establish the Schwarz Reflection Principle for $J$-complex discs attached to a real analytic $J$-totally real submanifold of an almost complex manifold with real analytic $J$. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in \calc^{k,\alpha}-classes.Comment: 21 page

Levi-flat filling of real two-spheres in symplectic manifolds (I)

Let (M,J,w) be a manifold with an almost complex structure J tamed by a symplectic form w. We suppose that M has complex dimension two, is Levi convex and has bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M may be foliated by the boundaries of pseudoholomorphic discs.Comment: 18 page