Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Properties of fixed point sets and a characterization of the ball in $ {\mathbb{C}}^n$


Authors: Buma L. Fridman and Daowei Ma
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
Published electronically: June 29, 2006
MathSciNet review: 2280191
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. E. Bedford and J. Dadok, Bounded domains with prescribed group of automorphisms, Comment. Math. Helv., 62 (1987), 561-572. MR 0920057 (89c:32078)
  • 2. 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.
  • 3. H. Cartan, Sur les fonctions de plusieurs variables complexes. L'itération des transformations intérieures d'un domaine borné, Math. Z., 35 (1932) 760-773. MR 1545327
  • 4. S. D. Fisher and John Franks, The fixed points of an analytic self-mapping, Proc. AMS, 99 (1987), 76-78. MR 0866433 (87m:30069)
  • 5. B. L. Fridman, K. T. Kim, S. G. Krantz, & D. Ma, On fixed points and determining sets for holomorphic automorphisms, Michigan Math. J., 50 (2002), 507-515. MR 1935150 (2003i:32034)
  • 6. 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.
  • 7. B. L. Fridman and E. A. Poletsky, Upper semicontinuity of automorphism groups, Math. Ann., 299 (1994), 615-628. MR 1286888 (96b:32040)
  • 8. B. L. Fridman, D. Ma, E. A. Poletsky, Upper semicontinuity of the dimensions of automorphism groups of domains in $ {\mathbb{C}}^n$, Amer. J. Math., 125 (2003), 289-299. MR 1963686 (2004f:32027)
  • 9. R. E. Greene and S. 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 0808832 (87b:32061)
  • 10. R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, 699, Springer, Berlin, 1979. MR 0521983 (81a:53002)
  • 11. D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, $ 2^{\rm nd}$ ed., Lecture Notes in Mathematics, v. 55, Springer-Verlag, New York, 1975. MR 0365399 (51:1651)
  • 12. K. T. Kim, S. G. Krantz, Determining sets and fixed points for holomorphic endomorphisms, Contemporary Math. 328 (2003), 239-246. MR 1990405 (2004e:32012)
  • 13. W. Klingenberg, Riemannian Geometry, $ 2^{\rm nd}$ ed., de Gruyter Studies in Mathematics, Berlin, 1995. MR 1330918 (95m:53003)
  • 14. S. Kobayashi, Hyperbolic Complex Spaces, Springer, 1999. MR 1635983 (99m:32026)
  • 15. K. Leschinger, Über fixpunkte holomorpher Automorphismen, Manuscripta Math., 25 (1978), 391-396. MR 0509592 (80e:55005)
  • 16. D. Ma, Upper semicontinuity of Isotropy and automorphism groups, Math. Ann., 292 (1992), 533-545. MR 1152949 (92m:32055)
  • 17. B. Maskit, The conformal group of a plane domain, Amer. J. Math., 90 (1968), 718-722. MR 0239078 (39:437)
  • 18. D. Montgomery, L. Zippin, Topological transformation groups, Interscience, New York, 1955. MR 0073104 (17:383b)
  • 19. 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. MR 0538091 (80h:30010)
  • 20. N. Suita, On fixed points of conformal self-mappings, Hokkaido Math. J., 10 (1981), 667-671. MR 0662329 (83f:30006)
  • 21. J.-P. Vigué, Sur les ensembles d'unicité pour les automorphismes analytiques d'un domaine borné, C. R. Acad. Sci. Paris, Ser. I 336 (2003), 589-592. MR 1981474 (2004b:32028)
  • 22. J.-P. Vigué, Ensembles d'unicité pour les automorphismes et les endomorphismes analytiques d'un domaine borné, Annales Institut Fourier, 55 (2005), 147-159. MR 2141692 (2006b:32023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32M05, 54H15

Retrieve articles in all journals with MSC (2000): 32M05, 54H15


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

DOI: https://doi.org/10.1090/S0002-9939-06-08641-2
Received by editor(s): August 2, 2005
Published electronically: June 29, 2006
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society