Characterization of the Hilbert ball by its automorphism group
Authors:
Kang-Tae Kim and Steven G. Krantz
Journal:
Trans. Amer. Math. Soc. 354 (2002), 2797-2818
MSC (2000):
Primary 32A07
DOI:
https://doi.org/10.1090/S0002-9947-02-02895-7
Published electronically:
February 12, 2002
MathSciNet review:
1895204
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $\Omega$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ânormal familiesâ arguments in the construction of the limit biholomorphism.
- Bedford, E. and Pinchuk, S., Domains in ${\mathbb {C}}^2$ with non-compact holomorphic automorphism group (translated from Russian), Math. USSR-Sb. 63(1989), 141â151.
- Eric Bedford and Sergey Pinchuk, Domains in ${\bf C}^{n+1}$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165â191. MR 1120679, DOI https://doi.org/10.1007/BF02921302
- Bedford, E. and Pinchuk, S., Convex domains with non-compact automorphism group (translated from Russian), Russian Acad. Sci. Sb. Math. 82(1995), 1â20.
- Eric Bedford and Sergey Pinchuk, Domains in ${\bf C}^2$ with noncompact automorphism groups, Indiana Univ. Math. J. 47 (1998), no. 1, 199â222. MR 1631557, DOI https://doi.org/10.1512/iumj.1998.47.1552
- François Berteloot, Sur certains domaines faiblement pseudoconvexes dont le groupe dâautomorphismes analytiques est non compact, Bull. Sci. Math. 114 (1990), no. 4, 411â420 (French, with English summary). MR 1077269
- François Berteloot, Un principe de localisation pour les domaines faiblement pseudoconvexes de ${\bf C}^2$ dont le groupe dâautomorphismes holomorphes nâest pas compact, AstĂ©risque 217 (1993), 5, 13â27 (French, with French summary). Colloque dâAnalyse Complexe et GĂ©omĂ©trie (Marseille, 1992). MR 1247748
- François Berteloot, Characterization of models in $\mathbf C^2$ by their automorphism groups, Internat. J. Math. 5 (1994), no. 5, 619â634. MR 1297410, DOI https://doi.org/10.1142/S0129167X94000322
- F. Berteloot and G. CĆurĂ©, Domaines de ${\bf C}^2$, pseudoconvexes et de type fini ayant un groupe non compact dâautomorphismes, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 77â86 (French, with English summary). MR 1112192
- Bochner, S. and Martin, W., Several complex variables, Princeton Univ. Press, 1948.
- Siqi Fu, A. V. Isaev, and S. G. Krantz, Examples of domains with non-compact automorphism groups, Math. Res. Lett. 3 (1996), no. 5, 609â617. MR 1418575, DOI https://doi.org/10.4310/MRL.1996.v3.n5.a4
- Siqi Fu, A. V. Isaev, and Steven G. Krantz, Reinhardt domains with non-compact automorphism groups, Math. Res. Lett. 3 (1996), no. 1, 109â122. MR 1393388, DOI https://doi.org/10.4310/MRL.1996.v3.n1.a11
- Sidney Frankel, Complex geometry of convex domains that cover varieties, Acta Math. 163 (1989), no. 1-2, 109â149. MR 1007621, DOI https://doi.org/10.1007/BF02392734
- Robert E. Greene and Steven G. Krantz, Characterizations of certain weakly pseudoconvex domains with noncompact automorphism groups, Complex analysis (University Park, Pa., 1986) Lecture Notes in Math., vol. 1268, Springer, Berlin, 1987, pp. 121â157. MR 907058, DOI https://doi.org/10.1007/BFb0097301
- A. V. Isaev and S. G. Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999), no. 1, 1â38. MR 1706680, DOI https://doi.org/10.1006/aima.1998.1821
- A. V. Isaev and S. G. Krantz, On the boundary orbit accumulation set for a domain with noncompact automorphism group, Michigan Math. J. 43 (1996), no. 3, 611â617. MR 1420595, DOI https://doi.org/10.1307/mmj/1029005546
- A. V. Isaev and S. G. Krantz, Finitely smooth Reinhardt domains with non-compact automorphism group, Illinois J. Math. 41 (1997), no. 3, 412â420. MR 1458181
- A. V. Isaev and S. G. Krantz, Hyperbolic Reinhardt domains in ${\bf C}^2$ with noncompact automorphism group, Pacific J. Math. 184 (1998), no. 1, 149â160. MR 1626532, DOI https://doi.org/10.2140/pjm.1998.184.149
- Kang-Tae Kim, Domains with noncompact automorphism groups, Recent developments in geometry (Los Angeles, CA, 1987) Contemp. Math., vol. 101, Amer. Math. Soc., Providence, RI, 1989, pp. 249â262. MR 1034985, DOI https://doi.org/10.1090/conm/101/1034985
- Kang-Tae Kim, Complete localization of domains with noncompact automorphism groups, Trans. Amer. Math. Soc. 319 (1990), no. 1, 139â153. MR 986028, DOI https://doi.org/10.1090/S0002-9947-1990-0986028-X
- Kang-Tae Kim, Domains in ${\bf C}^n$ with a piecewise Levi flat boundary which possess a noncompact automorphism group, Math. Ann. 292 (1992), no. 4, 575â586. MR 1157315, DOI https://doi.org/10.1007/BF01444637
- Kang-Tae Kim, Geometry of bounded domains and the scaling techniques in several complex variables, Lecture Notes Series, vol. 13, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1320264
- Kang-Tae Kim, On a boundary point repelling automorphism orbits, J. Math. Anal. Appl. 179 (1993), no. 2, 463â482. MR 1249831, DOI https://doi.org/10.1006/jmaa.1993.1362
- Kim, K.-T., Two examples for scaling methods in several complex variables, RIM-GARC Preprint Series, Seoul National University 95-53(1995).
- Kang-Tae Kim and Steven G. Krantz, A crash course in the function theory of several complex variables, Complex geometric analysis in Pohang (1997), Contemp. Math., vol. 222, Amer. Math. Soc., Providence, RI, 1999, pp. 3â37. MR 1653042, DOI https://doi.org/10.1090/conm/222/03172
- Steven G. Krantz, Function theory of several complex variables, 2nd ed., The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1162310
- Steven G. Krantz, Convexity in complex analysis, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 119â137. MR 1128519
- LĂĄszlĂł Lempert, La mĂ©trique de Kobayashi et la reprĂ©sentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427â474 (French, with English summary). MR 660145
- LĂĄszlĂł Lempert, The Dolbeault complex in infinite dimensions. I, J. Amer. Math. Soc. 11 (1998), no. 3, 485â520. MR 1603858, DOI https://doi.org/10.1090/S0894-0347-98-00266-5
- Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de MatemĂĄtica [Mathematical Notes], 107. MR 842435
- Raghavan Narasimhan, Several complex variables, The University of Chicago Press, Chicago, Ill.-London, 1971. Chicago Lectures in Mathematics. MR 0342725
- Rosay, J.-P., Sur une characterization de la boule parmi les domains de $\mathbb {C}^n$ par son groupe dâautomorphismes, Ann. Inst. Four. Grenoble XXIX(1979), 91â97.
- Wong, B., Characterization of the ball in $\mathbb {C}^n$ by its automorphism group, Invent. Math. 41(1977), 253â257.
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Additional Information
Kang-Tae Kim
Affiliation:
Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, The Republic of Korea
Email:
kimkt@postech.edu
Steven G. Krantz
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
MR Author ID:
106160
Email:
sk@math.wustl.edu
Received by editor(s):
January 20, 2000
Received by editor(s) in revised form:
March 23, 2001
Published electronically:
February 12, 2002
Article copyright:
© Copyright 2002
American Mathematical Society