Levi-flat filling of real two-spheres in symplectic manifolds. I. (Q650980)

The paper is to devoted to the proof of the following theorem: Let \((M, J, \omega)\) be an almost complex manifold of complex dimension \(2\) with a taming symplectic form with bounded geometry such that: a) there is no compact \(J\)-complex sphere in \(M\); b) the boundary \(\partial M\) of \(M\) is a smooth Levi-convex hypersurface not containing any germ of a non-constant \(J\)-holomorphic disc. If \(S^2\) is a real 2-sphere with two elliptic points embedded in \(\partial M\), there exists a unique smooth 1-parameter family of disjoint Bishop discs for \(S^2\), filling a real Levi-flat hypersurface \(\Gamma \subset M\) with boundary \(\partial \Gamma = S^2\), which is smooth up to the boundary except at the two elliptic points. This statement is close to other results, proved by several authors in various forms. The main purpose of this paper is however to provide a detailed and self-contained proof, to be used in a forthcoming paper devoted to the problem of filling 2-spheres with elliptic and hyperbolic points. The results and techniques considered here will allow in the second paper to focus on the behaviour of Bishop disks near hyperbolic points.