Extension of CR functions from boundaries in ${\mathbb C}^n \times {\mathbb R}$

TL;DR

This paper proves that smooth CR functions on certain bounded domains in complex space with real-analytic boundaries extend smoothly into the interior, and holomorphically if the boundary is real-analytic, with applications to the Levi-flat Plateau problem.

Contribution

It establishes extension results for CR functions from boundaries with elliptic CR singularities in complex spaces, generalizing classical Hartogs-Bochner theorems.

Findings

CR functions extend smoothly into the domain

Holomorphic extension if boundary is real-analytic

Application to Levi-flat Plateau problem

Abstract

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.