Browse > Article

DOMAINS WITH Ck CR CONTRACTIONS  

Kim, Sung-Yeon (Department of Mathematics Education Kangwon National University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.23, no.1, 2010 , pp. 11-27 More about this Journal
Abstract
Let $\Omega$ be a domain with smooth boundary in ${\mathbb{C}}^{n+1}$ and let $p{\in}{\partial}{\Omega}$. Suppose that $\Omega$ is Kobayashi hyperbolic and p is of Catlin multi-type ${\tau}=({\tau}_0,{\ldots},{\tau}_n)$. In this paper, we show that $\Omega$ admits a $C^{k}$ contraction at p with $k{\geq}\mid{\tau}\mid+1$ if and only if $\Omega$ is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial.
Keywords
CR contraction; weighted homogeneous domain;
Citations & Related Records
연도 인용수 순위
  • Reference
1 M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real Submanifolds in Complex Space and Their mappings, Princeton Math. Series 47, Princeton Univ. Press, New Jersey, 1999.
2 M. S. Baouendi, L. P. Rothschild and F. Treves CR structures with group action and extendability of CR functions, Invent. Math. 82 (1985), no. 2, 359-396.   DOI
3 F. Berteloot, Methodes de changement d'echelles en analyse complexe, A draft for lectures at C.I.R.M. (Luminy, France) in 2003.
4 D. W. Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), no. 3, 529-586.   DOI   ScienceOn
5 J. P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615-637.   DOI   ScienceOn
6 K. T. Kim and S. Y. Kim, CR hypersurfaces with a contracting automorphism, J. Geom. Anal. 18 (2008), no. 3, 800-834.   DOI   ScienceOn
7 K. T. Kim and J. C. Yoccoz, Real hypersurface with a holomorphic contraction, preprint.
8 S. Kobayashi, Hyperbolic complex spaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 318. Springer-Verlag, Berlin, 1998.
9 S. G. Krantz, Function theory of several complex variables, AMS Chelsea, Amer. Math. Soc. 1992.
10 J. P. Rosay, Sur une caracterisation de la boule parmi les domaines de $C^n$ par son groupe d'automorphismes. Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, ix, 91-97.   DOI
11 N. Tanaka, On the pseudoconformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397-429.   DOI
12 T. Ueda, Normal forms of attracting holomorphic maps, Math. J. of Toyama Univ. 22 (1999), 25-34.
13 B. Wong, Characterization of the unit ball in $C^n$ by its automorphism group. Invent. Math. 41 (1977), no. 3, 253-257.   DOI