Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T04:07:53.823Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains

Published online by Cambridge University Press:  20 November 2018

Steven Krantz
Steven Krantz*
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study and generalize a classical theoremof L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Keywords

32A3830H5032A1032M99Bers’s theoremalgebras of holomorphic functionsnoncompact automorphism groupbiholomorphic equivalence
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 346 - 353
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Bers, L., On rings of analytic functions. Bull. Amer. Math. Soc. 54(1948), 311-315. http://dx.doi.org/10.1090/S0002-9904-1948-08992-3 Google Scholar
[2] Fefferman, C., The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26(1974), 1–65. http://dx.doi.org/10.1007/BF01406845 Google Scholar
[3] Graham, C. R., Scalar boundary invariants and the Bergman kernel. Complex analysis, II (College Park, Md., 1985-86), Lecture Notes in Math., 1276, Springer, Berlin, 1987, pp. 108–135. http://dx.doi.org/10.1007/BFb0078958 Google Scholar
[4] Greene, R. E., Kim, K.-T., and Krantz, S. G., The geometry of complex domains.Progress in Mathematics, 291, BirkhâuserBoston, Boston, MA, 2011. http://dx.doi.org/10.1007/978-0-8176-4622-6 Google Scholar
[5] Greene, R. E. and Krantz, S. G., Deformations of complex structure, estimates for the d-equation, and stability of the Bergman kernel. Advances in Math. 43(1982), no. 1,1-86.http://dx.doi.org/10.1016/0001-8708(82)90028-7 http://dx.doi.org/10.101 6/0001-8708(82)90028-7 Google Scholar
[6] Kerzman, N. and Nagel, A., Finitely generated ideals in certain function algebras. J. Functional Analysis 7(1971), 212–215. http://dx.doi.org/10.1016/0022-1236(71)90053-X Google Scholar
[7] Klembeck, P. F., Kâhlermetrics of negative curvature, the Bergman metric near the boundary and the Kobayashi metric on smooth bounded strictly pseudoconvex sets. Indiana Univ. Math. J. 27(1978), no. 2, 275–282. http://dx.doi.org/10.1512/iumj.1978.27.27020 Google Scholar
[8] Krantz, S. G., Function theory of several complex variables. American Mathematical Society, Providence, RI, 2001.Google Scholar
[9] Krantz, S. G., Geometric function theory. Explorations in complex analysis. Cornerstones, Birkhâuser Boston, Boston, MA, 2006.Google Scholar
[10] Qi-Keng, L., On Kâhlermanifolds with constant curvature. Acta. Math.Sinica 16(1966), 269–281 (Chinese); Chinese Math. 9(1966), 283–298.Google Scholar
[11] Ohsawa, T., A remark on the completeness of the Bergman metric. Proc. Japan Acad. Ser. A Math. Sci. 57(1981), no. 4, 238–240. http://dx.doi.org/10.3792/pjaa.57.238 Google Scholar
[12] Siu, Y.-T., The d problem with uniform bounds on derivatives.Math. Ann. 207(1974), 163–176. http://dx.doi.org/10.1007/BF01362154 Google Scholar
[13] Wong, B., Characterizations of the ball in C” by its automorphism group. Invent. Math. 41(1977), no. 3, 253–257. http://dx.doi.org/10.1007/BF01403050 Google Scholar
[14] Zame, W. R., Homomorphisms of rings of germs of analytic functions. Proc. Amer. Math. Soc. 33(1972), 410–414. http://dx.doi.org/10.1090/S0002-9939-1972-0289808-6 Google Scholar
[15] Zame, W. R., Induced homomorphisms of algebras of analytic germs. Complex Analysis, 1972 (Proc. Conf, Rice Univ., Houston, Tex., 1972), Vol. II: Analysis on singularities. Rice Univ. Studies 59(1973), no. 2, 157–163.Google Scholar