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 neighbourhood 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 $$f :\partial \Omega \rightarrow {\mathbb {C}}$$ is a real analytic function that is CR at CR points of $$\partial \Omega $$ , then f extends to a holomorphic function on a neighbourhood of $${\overline{\Omega }}$$ in $${\mathbb {C}}^n \times {\mathbb {C}}$$ .