Properties of fixed point sets and a characterization of the ball in ${\mathbb C}^n$
HTML articles powered by AMS MathViewer
- by Buma L. Fridman and Daowei Ma PDF
- Proc. Amer. Math. Soc. 135 (2007), 229-236 Request permission
Abstract:
We study the fixed point sets of holomorphic self-maps of a bounded domain in ${\mathbb C}^n$. Specifically we investigate the least number of fixed points in general position in the domain that forces any automorphism (or endomorphism) to be the identity. We have discovered that in terms of this number one can give the necessary and sufficient condition for the domain to be biholomorphic to the unit ball. Other theorems and examples generalize and complement previous results in this area, especially the recent work of Jean-Pierre Vigué.References
- Eric Bedford and Jiri Dadok, Bounded domains with prescribed group of automorphisms, Comment. Math. Helv. 62 (1987), no. 4, 561–572. MR 920057, DOI 10.1007/BF02564462
- H. Cartan, Les fonctions de deux variables complexeses et le problème de la représentation analytique, J. Math. pures et appl., 9$^e$ série, 11 (1931) 1-114.
- Henri Cartan, Sur les fonctions de plusieurs variables complexes. L’itération des transformations intérieures d’un domaine borné, Math. Z. 35 (1932), no. 1, 760–773 (French). MR 1545327, DOI 10.1007/BF01186587
- S. D. Fisher and John Franks, The fixed points of an analytic self-mapping, Proc. Amer. Math. Soc. 99 (1987), no. 1, 76–78. MR 866433, DOI 10.1090/S0002-9939-1987-0866433-8
- B. L. Fridman, K. T. Kim, S. G. Krantz, and D. Ma, On fixed points and determining sets for holomorphic automorphisms, Michigan Math. J. 50 (2002), no. 3, 507–515. MR 1935150, DOI 10.1307/mmj/1039029980
- B. L. Fridman, K. T. Kim, S. G. Krantz, & D. Ma, On determining sets for holomorphic automorphisms, to appear in Rocky Mountain J. of Math.
- Buma L. Fridman and Evgeny A. Poletsky, Upper semicontinuity of automorphism groups, Math. Ann. 299 (1994), no. 4, 615–628. MR 1286888, DOI 10.1007/BF01459802
- Buma L. Fridman, Daowei Ma, and Evgeny A. Poletsky, Upper semicontinuity of the dimensions of automorphism groups of domains in ${\Bbb C}^N$, Amer. J. Math. 125 (2003), no. 2, 289–299. MR 1963686
- Robert E. Greene and Steven G. Krantz, Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, Indiana Univ. Math. J. 34 (1985), no. 4, 865–879. MR 808832, DOI 10.1512/iumj.1985.34.34048
- R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, Vol. 55, Springer-Verlag, Berlin-New York, 1975 (German). Zweite Auflage. MR 0365399
- Kang-Tae Kim and Steven G. Krantz, Determining sets and fixed points for holomorphic endomorphisms, Function spaces (Edwardsville, IL, 2002) Contemp. Math., vol. 328, Amer. Math. Soc., Providence, RI, 2003, pp. 239–246. MR 1990405, DOI 10.1090/conm/328/05785
- Wilhelm P. A. Klingenberg, Riemannian geometry, 2nd ed., De Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin, 1995. MR 1330918, DOI 10.1515/9783110905120
- Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
- Karl Leschinger, Über Fixpunkte holomorpher Automorphismen, Manuscripta Math. 25 (1978), no. 4, 391–396 (German, with English summary). MR 509592, DOI 10.1007/BF01168050
- Daowei Ma, Upper semicontinuity of isotropy and automorphism groups, Math. Ann. 292 (1992), no. 3, 533–545. MR 1152949, DOI 10.1007/BF01444634
- Bernard Maskit, The conformal group of a plane domain, Amer. J. Math. 90 (1968), 718–722. MR 239078, DOI 10.2307/2373479
- Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
- Ernst Peschl and Matti Lehtinen, A conformal self-map which fixes three points is the identity, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 85–86. MR 538091, DOI 10.5186/aasfm.1978-79.0419
- Nobuyuki Suita, On fixed points of conformal self-mappings, Hokkaido Math. J. 10 (1981), no. Special Issue, 667–671. MR 662329
- Jean-Pierre Vigué, Sur les ensembles d’unicité pour les automorphismes analytiques d’un domaine borné, C. R. Math. Acad. Sci. Paris 336 (2003), no. 7, 589–592 (French, with English and French summaries). MR 1981474, DOI 10.1016/S1631-073X(03)00137-7
- Jean-Pierre Vigué, Ensembles d’unicité pour les automorphismes et les endomorphismes analytiques d’un domaine borné, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 147–159 (French, with English and French summaries). MR 2141692
Additional Information
- Buma L. Fridman
- Affiliation: Department of Mathematics, Wichita State University, Wichita, Kansas 67260-0033
- Email: buma.fridman@wichita.edu
- Daowei Ma
- Affiliation: Department of Mathematics, Wichita State University, Wichita, Kansas 67260-0033
- Email: dma@math.wichita.edu
- Received by editor(s): August 2, 2005
- Published electronically: June 29, 2006
- Communicated by: Mei-Chi Shaw
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 229-236
- MSC (2000): Primary 32M05, 54H15
- DOI: https://doi.org/10.1090/S0002-9939-06-08641-2
- MathSciNet review: 2280191