Codimension two CR singular submanifolds and extensions of CR functions

Abstract

Let $M\subset {\mathbb{C}}^{n+1}$, $n\ge 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $\Omega \subset {\mathbb{C}}^n\times \mathbb{R}$, $n\ge 2$, be a bounded domain with a connected real-analytic boundary such that $\partial \Omega$ has only nondegenerate CR singularities. We prove that if $f:\partial \Omega \to \mathbb{C}$ is a real-analytic function that is CR at CR points of $\partial \Omega$, then $f$ extends to a holomorphic function on a neighborhood of $\overline{\Omega }$ in ${\mathbb{C}}^n\times \mathbb{C}$.

16 pages, 1 figure, fixed typos, updated references. To appear in Journal of Geometric Analysis

References in corpus (4)

• Normal forms for CR singular codimension two Levi-flat submanifolds
• On complex points of codimension 2 submanifolds
• Extension of CR functions from boundaries in ${\mathbb{C}}^n\times \mathbb{R}$
• Flattening of CR singular points and analyticity of local hull of holomorphy

Cited by in corpus (5)

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