The Journal of Geometric Analysis

, Volume 27, Issue 3, pp 2453–2471 | Cite as

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Article

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 \({\mathbb {C}}^n \times {\mathbb {C}}\).

Keywords

Extension of CR functions Hartogs–Bochner CR singularity Levi-flat Plateau problem 

Mathematics Subject Classification

Primary 32V40 Secondary 32V25 

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Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA
  2. 2.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

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