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Local equivalence problem for Levi flat hypersurfaces
Author:
Giuseppe Della Sala
Journal:
Proc. Amer. Math. Soc. 139 (2011), 2449-2461
MSC (2010):
Primary 32V40
Posted:
December 3, 2010
MathSciNet review:
2784811
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Abstract |
References |
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Additional Information
Abstract: In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence:  if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows us to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of non-equivalent hypersurfaces which are almost everywhere analytic.
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- Aldo Andreotti and Gregory A. Fredricks, Embeddability of real analytic Cauchy-Riemann manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 2, 285-304. MR 541450 (80h:32019)
- [2]
- M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103 (2000b:32066)
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- Eric Bedford and Paolo De Bartolomeis, Levi flat hypersurfaces which are not holomorphically flat, Proc. Amer. Math. Soc. 81 (1981), no. 4, 575-578. MR 601733 (82a:32025)
- [4]
- Steven R. Bell, The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1228442 (94k:30013)
- [5]
- Steve Bell and László Lempert, A
 Schwarz reflection principle in one and several complex variables, J. Differential Geom. 32 (1990), no. 3, 899-915. MR 1078165 (91k:32017)
- [6]
- Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes. I, Ann. Mat. Pura Appl., IV. Ser. 11 (1932), 17-90 (French).
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- Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333-354 (French). MR 1556687
- [8]
- S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
- [9]
- F. Reese Harvey and H. Blaine Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223-290. MR 0425173 (54:13130)
- [10]
- Martin Kolář, Normal forms for hypersurfaces of finite type in
 , Math. Res. Lett. 12 (2005), no. 5-6, 897-910. MR 2189248 (2007d:32034)
- [11]
- Henri Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo 23 (1907), 185-220.
- [12]
- Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of
 complex variables, J. Math. Soc. Japan 14 (1962), 397-429. MR 0145555 (26:3086)
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Additional Information
Giuseppe Della Sala
Affiliation:
Fakultät für Mathematik, Nordbergstrasse 15, University of Vienna, 1090 Vienna, Austria
Email:
giuseppe.dellasala@univie.ac.at, beppe.dellasala@gmail.com
DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10746-3
PII:
S 0002-9939(2010)10746-3
Keywords:
Levi flat hypersurface,
equivalence problem
Received by editor(s):
March 1, 2010
Received by editor(s) in revised form:
June 17, 2010
Posted:
December 3, 2010
Additional Notes:
This paper was written while the author was a post-doc at the University of Vienna, supported by BMWF grant Y377. He is very grateful to B. Lamel for posing the question in the first place, for several conversations regarding the problem and for otherwise helping in many crucial ways. He also wishes to thank A. Saracco for discussing the topic and for a useful suggestion about section 3, and F. Forstneric for pointing out reference \cite{BD}.
Communicated by:
Franc Forstneric
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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