Codimension two CR singular submanifolds and extensions of CR functions

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 ${\mathbb{C}}^n \times {\mathbb{C}}$.