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

TL;DR

This paper proves that under certain conditions, a real two-sphere with two elliptic points embedded in the boundary of a Levi convex symplectic manifold can be foliated by pseudoholomorphic discs, advancing the understanding of Levi-flat fillings.

Contribution

It establishes the existence of Levi-flat fillings of real two-spheres with elliptic points in Levi convex symplectic manifolds, extending previous results to new geometric settings.

Findings

Real two-spheres with two elliptic points can be foliated by pseudoholomorphic discs.

The result applies to manifolds with bounded geometry and Levi convexity.

The foliation provides a new tool for studying the boundary behavior of pseudoholomorphic curves.

Abstract

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.