Codimension two CR singular submanifolds and extensions of CR functions

TL;DR

This paper proves that real-analytic CR functions on certain codimension two CR singular submanifolds extend holomorphically, extending classical results to more complex geometric settings with CR singularities.

Contribution

It establishes holomorphic extension of CR functions on nondegenerate, holomorphically flat CR singular submanifolds, generalizing the Hartogs-Bochner theorem to new geometric contexts.

Findings

CR functions extend holomorphically near CR singularities

Extension results apply to bounded domains with CR singular boundaries

Generalizes classical extension theorems to codimension two CR singular manifolds

Abstract

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, 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 $\Omega \subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial \Omega$ has only nondegenerate CR singularities. We prove that if $f \colon \partial \Omega \to {\mathbb{C}}$ 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 ${\mathbb{C}}^n \times {\mathbb{C}}$.