Singular set of a Levi-flat hypersurface is Levi-flat
- 179 Downloads
We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.
Mathematics Subject Classification (2000)32C07 32V40 37F75 32B20 14P15
The author would like to acknowledge Peter Ebenfelt for suggesting the study of this problem when the author was still a graduate student, and also for many conversations on the topic. The author would also like to thank Xianghong Gong, John P. D’Angelo, Salah Baouendi, Linda Rothschild, and Arturo Fernández-Pérez for fruitful discussions on topics related to this research and suggestions on the manuscript.
- 2.Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real submanifolds in complex space and their mappings. In: Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)Google Scholar
- 9.D’Angelo, J.P.: Several complex variables and the geometry of real hypersurfaces. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)Google Scholar
- 10.Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math. (2) 102(2), 384–371 (1978)Google Scholar
- 14.Krantz, S.G.: Function theory of several complex variables. AMS Chelsea Publishing, Providence (2001) (Reprint of the 1992 edition)Google Scholar
- 16.Lebl, J.: Singularities and Complexity in CR Geometry, Ph.D. thesis, University of California at San Diego (2007)Google Scholar
- 19.Siu, Y.-T.: Techniques of extension of analytic objects. In: Lecture Notes in Pure and Applied Mathematics, vol. 8. Marcel Dekker Inc., New York (1974)Google Scholar