On the Levi-flat Plateau problem

TL;DR

This paper addresses the Levi-flat Plateau problem for certain real-analytic submanifolds with CR singularities, establishing existence and uniqueness of Levi-flat hypersurfaces with prescribed boundaries in complex Euclidean spaces.

Contribution

It provides a solution to the Levi-flat Plateau problem under specific conditions involving CR singularities and real-analytic maps, extending previous results in complex analysis.

Findings

Existence of a unique Levi-flat hypersurface with given boundary.

Regularity results for CR automorphisms of domains in complex spaces.

Characterization of boundary behavior in the presence of CR singularities.

Abstract

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.