Boundary problem for Levi flat graphs

TL;DR

This paper proves that for a specific class of boundary graphs over convex domains in complex space, the Levi-flat hypersurface they bound is necessarily smooth and non-singular, extending previous general conditions.

Contribution

It establishes that when the boundary is a graph over a strongly convex domain, the Levi-flat hypersurface is also a smooth graph, ensuring non-singularity in this case.

Findings

Levi-flat hypersurface is a smooth graph over the domain

Hypersurface is necessarily non-singular

Extends previous conditions to specific boundary graphs

Abstract

In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$ bounded by $S$. In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $\Omega\subset C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a smooth function over $\Omega$. In particular, $M$ is non-singular.