We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in and to give a sufficient condition for the complete hyperbolicity of a domain in .
Let be a manifold with an almost complex structure tamed by a symplectic form . We suppose that has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of can be foliated by the boundaries of pseudoholomorphic discs.
The paper discusses some aspects of Gromov’s theory of gluing complex discs to Lagrangian manifolds.
We consider a compact almost complex manifold with smooth Levi convex boundary and a symplectic tame form . Suppose that is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into . We prove a result on filling by holomorphic discs.
We establish the Schwarz Reflection Principle for -complex discs attached to a real analytic -totally real submanifold of an almost complex manifold with real analytic . We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in -classes.
We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.
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