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A hypersurface in $\mathbb{C}^2$ whose stability group is not determined by $2$-jets

Author: R. Travis Kowalski
Journal: Proc. Amer. Math. Soc. 130 (2002), 3679-3686
MSC (2000): Primary 32H12, 32V20
Published electronically: May 15, 2002
MathSciNet review: 1920048
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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 $\mathbb{C}^2$.

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  • [BER00] 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 (electronic). MR 1754643, 10.1090/S0273-0979-00-00863-6
  • [BG77] Thomas Bloom and Ian Graham, On “type” conditions for generic real submanifolds of 𝐶ⁿ, Invent. Math. 40 (1977), no. 3, 217–243. MR 0589930
  • [CM74] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 0425155
  • [ELZ00] 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:, (2000).
  • [Koh72] J. J. Kohn, Boundary behavior of 𝛿 on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. MR 0322365
  • [Kow01] R. T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in $\mathbb{C}^2$, E-print:, (2001).
  • [Poi07] H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo, II. Ser. 23 (1907), 544-547.
  • [Vit90] Several complex variables. I, Encyclopaedia of Mathematical Sciences, vol. 7, Springer-Verlag, Berlin, 1990. Introduction to complex analysis; A translation of Sovremennye problemy matematiki. Fundamentalnye 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

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Additional Information

R. Travis Kowalski
Affiliation: Department of Mathematics, 0112, University of California, San Diego, La Jolla, California 92093-0112

Received by editor(s): July 31, 2001
Published electronically: May 15, 2002
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2002 American Mathematical Society