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Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

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Abstract

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.

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References

  1. Bedford, E. Holomorphic continuation of smooth functions over Levi-flat hypersurfaces,Trans. Amer. Math. Soc. 232, 323–341, (1977).

    Article  MATH  MathSciNet  Google Scholar 

  2. Baouendi, M. S., Ebenfelt, P., and Rothschild, L. P. CR automorphisms of real analytic manifolds in complex space,Comm. Anal. Geom. 6(2), 291–315, (1998).

    MATH  MathSciNet  Google Scholar 

  3. Baouendi, M. S., Ebenfelt, P., and Rothschild, L. P.Real Submanifolds in Complex Space and Their Mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999.

    MATH  Google Scholar 

  4. Baouendi, M. S., Ebenfelt, P., and Rothschild, L. P. Local geometric properties of real submanifolds in complex space,Bull. Amer. Math. Soc. (N.S.) 37(3), 309–336, (2000).

    Article  MATH  MathSciNet  Google Scholar 

  5. Burns, D. and Gong, X. Singular Levi-flat real analytic hypersurfaces,Amer. J. Math. 121(1), 23–53, (1999).

    Article  MATH  MathSciNet  Google Scholar 

  6. Bierstone, E. and Milman, P. D. Semianalytic and subanalytic sets,Publ. Math. Inst. Hautes Etudes Sci. 67, 5–42, (1988).

    Article  MATH  MathSciNet  Google Scholar 

  7. Huang, X., Ji, S., and Yau, S. S. T. An example of a real analytic strongly pseudoconvex hypersurface which is not holomorphically equivalent to any algebraic hypersurface,Ark. Mat. 39(1), 75–93, (2001).

    Article  MATH  MathSciNet  Google Scholar 

  8. Łojasiewicz, S.Introduction to Complex Analytic Geometry, Birkhäuser Verlag, Basel, Translated from Polish by Maciej Klimek, 1991.

    MATH  Google Scholar 

  9. Stanton, N. K. Infinitesimal CR automorphisms of real hypersurfaces,Amer. J. Math. 118(1), 209–233, (1996).

    Article  MATH  MathSciNet  Google Scholar 

  10. Whitney, H.Complex Analytic Varieties, Addison-Wesley Publishing Co, Reading, Mass.-London-Don Mills, Ont., 1972.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, 92093-0112, La Jolla, CA

    Jiří Lebl

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  1. Jiří Lebl
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Correspondence to Jiří Lebl.

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Communicated by Steven Bell

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Lebl, J. Nowhere minimal CR submanifolds and Levi-flat hypersurfaces. J Geom Anal 17, 321–341 (2007). https://doi.org/10.1007/BF02930726

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  • DOI: https://doi.org/10.1007/BF02930726

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