Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate real hypersurfaces in $ \mathbb{C}^2$ 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

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce new biholomorphic invariants for real-analytic hypersurfaces in $ \mathbb{C}^2$ 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.

References [Enhancements On Off] (What's this?)

  • 1. M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math. 1 (1997), no. 1, 1-16. MR 99b:32022
  • 2. -, Real submanifolds in complex space and their mappings, Princeton University Press, Princeton, NJ, 1999. MR 1668103
  • 3. -, Convergence and finite determination of formal CR mappings, J. Amer. Math. Soc. 13 (2000), no. 4, 697-723 (electronic). MR 2001h:32063
  • 4. P. 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 (2004g:32033)
  • 6. M. Kolář, Normal forms for hypersurfaces of finite type in $ \mathbb{C}\sp 2$, Math. Res. Lett. 12 (2005), no. 5-6, 897-910. MR 2189248
  • 7. R. T. Kowalski, A hypersurface in $ \mathbb{C}^2$ whose stability group is not determined by 2-jets, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3679-3686 (electronic). MR 2003f:32018
  • 8. H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo II. Ser. 23 (1907), 185-220.
  • 9. N. 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 (2001h:32065)
  • 10. D. 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 (2004i:32056)

Similar Articles

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

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

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

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

American Mathematical Society