Degenerate real hypersurfaces in with few automorphisms

Authors:
Peter Ebenfelt, Bernhard Lamel and Dmitri Zaitsev

Journal:
Trans. Amer. Math. Soc. **361** (2009), 3241-3267

MSC (2000):
Primary 32H02

Published electronically:
January 28, 2009

MathSciNet review:
2485425

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce new biholomorphic invariants for real-analytic hypersurfaces in and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained.

**1.**M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild,*Parametrization of local biholomorphisms of real analytic hypersurfaces*, Asian J. Math.**1**(1997), no. 1, 1–16. MR**1480988**, 10.4310/AJM.1997.v1.n1.a1**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****3.**M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild,*Convergence and finite determination of formal CR mappings*, J. Amer. Math. Soc.**13**(2000), no. 4, 697–723 (electronic). MR**1775734**, 10.1090/S0894-0347-00-00343-X**4.**Peter Ebenfelt,*New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy*, J. Differential Geom.**50**(1998), no. 2, 207–247. MR**1684982****5.**P. Ebenfelt, B. Lamel, and D. Zaitsev,*Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case*, Geom. Funct. Anal.**13**(2003), no. 3, 546–573. MR**1995799**, 10.1007/s00039-003-0422-y**6.**Martin Kolář,*Normal forms for hypersurfaces of finite type in ℂ²*, Math. Res. Lett.**12**(2005), no. 5-6, 897–910. MR**2189248**, 10.4310/MRL.2005.v12.n6.a10**7.**R. Travis Kowalski,*A hypersurface in ℂ² whose stability group is not determined by 2-jets*, Proc. Amer. Math. Soc.**130**(2002), no. 12, 3679–3686 (electronic). MR**1920048**, 10.1090/S0002-9939-02-06545-0**8.**H. Poincaré,*Les fonctions analytiques de deux variables et la représentation conforme*, Rend. Circ. Mat. Palermo**II. Ser. 23**(1907), 185-220.**9.**Nancy K. Stanton,*Real hypersurfaces with no infinitesimal CR automorphisms*, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 469–473. MR**1771288**, 10.1090/conm/251/03888**10.**Dmitri Zaitsev,*Unique determination of local CR-maps by their jets: a survey*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.**13**(2002), no. 3-4, 295–305. Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001). MR**1984108**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
32H02

Retrieve articles in all journals with MSC (2000): 32H02

Additional Information

**Peter Ebenfelt**

Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Email:
pebenfel@math.ucsd.edu

**Bernhard Lamel**

Affiliation:
Fakultät für Mathematik, Universität Wien, A-1090 Wien, Austria

Email:
lamelb@member.ams.org

**Dmitri Zaitsev**

Affiliation:
School of Mathematics, Trinity College, Dublin 2, Ireland

Email:
zaitsev@maths.tcd.ie

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04626-1

Keywords:
Infinite type hypersurface,
normal form,
stability group

Received by editor(s):
December 7, 2006

Received by editor(s) in revised form:
July 31, 2007

Published electronically:
January 28, 2009

Additional Notes:
The first author was supported in part by NSF grants DMS-0100110 and DMS-0401215.

The second author was supported by the ANACOGA network and the FWF, Projekt P17111

The third author was supported in part by the RCBS Grant of the Trinity College Dublin. This publication has emanated from research conducted with the financial support of the Science Foundation Ireland

Article copyright:
© Copyright 2009
American Mathematical Society