A hypersurface in $\mathbb {C}^2$ whose stability group is not determined by $2$-jets
HTML articles powered by AMS MathViewer
- by R. Travis Kowalski
- Proc. Amer. Math. Soc. 130 (2002), 3679-3686
- DOI: https://doi.org/10.1090/S0002-9939-02-06545-0
- Published electronically: May 15, 2002
- PDF | Request permission
Abstract:
We give an example of a hypersurface in $\mathbb {C}^2$ through $0$ whose stability group at $0$ is determined by $3$-jets, but not by jets of any lesser order. We also examine some of the properties which the stability group of this infinite type hypersurface shares with the $3$-sphere in $\mathbb {C}^2$.References
- M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Local geometric properties of real submanifolds in complex space, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309–336. MR 1754643, DOI 10.1090/S0273-0979-00-00863-6
- Thomas Bloom and Ian Graham, On “type” conditions for generic real submanifolds of $\textbf {C}^{n}$, Invent. Math. 40 (1977), no. 3, 217–243. MR 589930, DOI 10.1007/BF01425740
- S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
- 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, E-print: http://arXiv.org/abs/math.CV/0107013, (2000).
- J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
- R. T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in $\mathbb {C}^2$, E-print: http://arXiv.org/abs/math.CV/0108165, (2001).
- H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo, II. Ser. 23 (1907), 544–547.
- Several complex variables. I, Encyclopaedia of Mathematical Sciences, vol. 7, Springer-Verlag, Berlin, 1990. Introduction to complex analysis; A translation of Sovremennye problemy matematiki. Fundamental′nye napravleniya, Tom 7, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985 [ MR0850489 (87f:32003)]; Translation by P. M. Gauthier; Translation edited by A. G. Vitushkin. MR 1043689
Bibliographic Information
- R. Travis Kowalski
- Affiliation: Department of Mathematics, 0112, University of California, San Diego, La Jolla, California 92093-0112
- Email: kowalski@math.ucsd.edu
- Received by editor(s): July 31, 2001
- Published electronically: May 15, 2002
- Communicated by: Mei-Chi Shaw
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3679-3686
- MSC (2000): Primary 32H12, 32V20
- DOI: https://doi.org/10.1090/S0002-9939-02-06545-0
- MathSciNet review: 1920048