Browse > Article
http://dx.doi.org/10.4134/JKMS.2003.40.3.487

PERTURBATION OF DOMAINS AND AUTOMORPHISM GROUPS  

Fridman, Buma L. (Department of Mathematics Wichita State University)
Ma, Daowei (Department of Mathematics Wichita State University)
Publication Information
Journal of the Korean Mathematical Society / v.40, no.3, 2003 , pp. 487-501 More about this Journal
Abstract
The paper is devoted to the description of changes of the structure of the holomorphic automorphism group of a bounded domain in \mathbb{C}^n under small perturbation of this domain in the Hausdorff metric. We consider a number of examples when an arbitrary small perturbation can lead to a domain with a larger group, present theorems concerning upper semicontinuity property of some invariants of automorphism groups. We also prove that the dimension of an abelian subgroup of the automorphism group of a bounded domain in \mathbb{C}^n does not exceed n.
Keywords
automorphism groups; perturbation of domains; Hausdorff distance; abelian subgroups;
Citations & Related Records

Times Cited By Web Of Science : 1  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
연도 인용수 순위
1 R. Greene and S. G. Krantz, The Automorphism groups of strongly pseudoconvex domains, Math. Ann. 261 (1982), 425–446.   DOI
2 R. Greene and S. G. Krantz, Stability of the Caratheodory and Kobayashi metrics and applications to biholomorphic mappings, Proceedings of Symposia in Pure Math. Providence: AMS 41 (1984), 77–94.
3 R. Greene and S. G. Krantz, Normal Families and the Semicontinuity of Isometry and Automorphism Groups, Math. Z. 190 (1985), 455–467.   DOI
4 K. Grove and H. Karcher, How to conjugate $C^1$-close group actions, Math. Z. 132 (1973), 11–20.   DOI
5 Sh. Kobayashi, Hyperbolic Complex Spaces, Springer-Verlag, 1998
6 D. Ma, Upper semicontinuity of isotropy and automorphism groups, Math. Ann. 292 (1992), 533–545.   DOI
7 D. Montgomery, L. Zippin, Topological transformation groups, Interscience, New York, 1955
8 R. Palais, Equivalence of nearby differentiable actions of a group, Bull. Amer. Math. Soc. 67 (1961), 362–364   DOI
9 E. Peschl and M. Lehtinen, A conformal self-map which fixes 3 points is the identity, Ann. Acad. Sci. Fenn., Ser. A I Math. 4 (1979), no. 1, 85–86.
10 R. Saerens and W. R. Zame, The isometry groups of manifolds and the automor-phism groups of domains, Trans. Amer. Math. Soc. 301 (1987), 413–429   DOI   ScienceOn
11 A. E. Tumanov and G. B. Shabat, Realization of linear Lie groups by biholomor-phic automorphisms of bounded domains, Funct. Anal. Appl. (1990), 255–257.   DOI
12 B. L. Fridman and E. A. Poletsky, Upper semicontinuity of automorphism groups, Math. Ann. 299 (1994), 615–628.   DOI
13 E. Bedford and J. Dadok, Bounded domains with prescribed group of automor-phisms, Comment. Math. Helv. 62 (1987), 561–572   DOI
14 G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972
15 D. Ebin, The manifold of Riemannian metrics, Global analysis, Proceedings of Symposium in Pure Mathematics, XV, AMS (1970), 17–40.
16 B. L. Fridman, K. T. Kim, S. G. Krantz and D. Ma, On fixed points and determin-ing sets for holomorphic automorphisms, Michigan Math. J. 50 (2002), 507–515.   DOI
17 B. L. Fridman, D. Ma and E. A. Poletsky, Upper semicontinuity of the dimensions of automorphism groups in $C^n$, to appear in Amer. J. Math 125 (2003)
18 B. L. Fridman, Biholomorphic invariants of a hyperbolic manifold and some applications, Trans. Amer. Math. Soc. 276 (1983), no. 2, 685–698.   DOI   ScienceOn
19 B. L. Fridman, A universal exhausting domain, Proc. Amer. Math. Soc. 98 (1986), 267–270.   DOI   ScienceOn