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

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 $f:\partial \Omega \to \mathbb{C}$ 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.

20 pages, to appear in Indiana Univ. Math. J; fixed several typos

References in corpus (2)

• Normal forms for CR singular codimension two Levi-flat submanifolds
• Flattening of CR singular points and analyticity of local hull of holomorphy

Cited by in corpus (7)

• Codimension two CR singular submanifolds and extensions of CR functions
• On Lewy extension for smooth hypersurfaces in ${\mathbb{C}}^n\times \mathbb{R}$
• On the Levi-flat Plateau problem
• A CR singular analogue of Severi's theorem
• Flattening a non-degenerate CR singular point of real codimension two
• Segre nondegenerate totally real subvarieties
• CR functions at CR singularities: approximation, extension, and hulls