The Journal of Geometric Analysis

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

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

  • Jiří Lebl
  • Alan Noell
  • Sivaguru Ravisankar


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


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

Mathematics Subject Classification

Primary 32V40 Secondary 32V25 



The first author was in part supported by NSF Grant DMS-1362337 and Oklahoma State University’s DIG and ASR grants.


  1. 1.
    Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bochner, S.: Green’s Formula and Analytic Continuation, Contributions to the Theory of Partial Differential Equations. Annals of Mathematics Studies, vol. 33, pp. 1–14. Princeton University Press, Princeton (1954)zbMATHGoogle Scholar
  3. 3.
    Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brown, A.B.: On certain analytic continuations and analytic homeomorphisms. Duke Math. J. 2(1), 20–28 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burcea, V.: A normal form for a real 2-codimensional submanifold in \(\mathbb{C}^{N+1}\) near a CR singularity. Adv. Math. 243, 262–295 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burcea, V.: On a family of analytic discs attached to a real submanifold \(M\subset {{\mathbb{C}}}^{N+1}\). Methods Appl. Anal. 20(1), 69–78 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Coffman, A.: CR singularities of real fourfolds in \({\mathbb{C}}^3\). Ill. J. Math. 53(3), 939–981 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dolbeault, P., Tomassini, G., Zaitsev, D.: On boundaries of Levi-flat hypersurfaces in \({\mathbb{C}}^n\). C. R. Math. Acad. Sci. Paris 341(6), 343–348 (2005). (English, with English and French summaries)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dolbeault, P., Tomassini, G., Zaitsev, D.: Boundary problem for Levi flat graphs. Indiana Univ. Math. J. 60(1), 161–170 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gong, X., Lebl, J.: Normal forms for CR singular codimension-two Levi-flat submanifolds. Pac. J. Math. 275(1), 115–165 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harris, G.A.: The traces of holomorphic functions on real submanifolds. Trans. Am. Math. Soc. 242, 205–223 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Henkin, G., Michel, V.: Principe de Hartogs dans les variétés CR. J. Math. Pures Appl. 81(12), 1313–1395 (2002). (French, with English and French summaries)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huang, X., Krantz, S.G.: On a problem of Moser. Duke Math. J 78(1), 213–228 (1995). doi: 10.1215/S0012-7094-95-07809-0 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huang, X., Yin, W.: A Bishop surface with a vanishing Bishop invariant. Invent. Math. 176(3), 461–520 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huang, X., Yin, W.: A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric. Int. Math. Res. Not. 15, 2789–2828 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Huang, X., Yin, W.: Flattening of CR singular points and analyticity of the local hull of holomorphy I. Math. Ann. 365(1–2), 381–399 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Huang, X., Yin, W.: Flattening of CR singular points and analyticity of local hull of holomorphy II (preprint). arXiv:1210.5146
  18. 18.
    Kenig, C.E., Webster, S.M.: The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67(1), 1–21 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lebl, J., Minor, A., Shroff, R., Son, D., Zhang, Y.: CR singular images of generic submanifolds under holomorphic maps. Ark. Mat. 52(2), 301–327 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lebl, J., Noell, A., Ravisankar, S.: Extension of CR functions from boundaries in \({\bf C}^n\times {\bf R}\). Indiana Univ. Math. J. (to appear). arXiv:1505.05255
  21. 21.
    Moser, J.K.: Analytic surfaces in \({\bf C}^2\) and their local hull of holomorphy. Ann. Acad. Sci. Fenn. Ser. A I Math 10, 397–410 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Moser, J.K., Webster, S.M.: Normal forms for real surfaces in \({ C}^{2}\) near complex tangents and hyperbolic surface transformations. Acta Math. 150(3–4), 255–296 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Severi, F.: Una proprietà fondamentale dei campi di olomorfismo di una variabile reale e di una variabile complessa. Atti della Reale Accademia Nazionale dei Lincei, Rome, Rendiconti 15, 487–490 (1932)zbMATHGoogle Scholar
  24. 24.
    Shabat, B.V.: Introduction to complex analysis. Part II, Translations of Mathematical Monographs, vol. 110, American Mathematical Society, Providence, RI, 1992. Functions of several variables; Translated from the third (1985) Russian edition by J. S. JoelGoogle Scholar
  25. 25.
    Slapar, M.: On complex points of codimension 2 submanifolds. J. Geom. Anal. 26(1), 206–219 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Whitney, H.: Complex Analytic Varieties. Addison-Wesley Publishing Co., Reading (1972)zbMATHGoogle Scholar

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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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