Characterization of the Hilbert ball by its automorphism group

Kim, Kang-Tae; Krantz, Steven

doi:10.1090/S0002-9947-02-02895-7

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: 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 ``normal families'' arguments in the construction of the limit biholomorphism.

[BP1] Bedford, E. and Pinchuk, S., Domains in ${{\mathbb C}}^2$ with non-compact holomorphic automorphism group (translated from Russian), Math. USSR-Sb. 63(1989), 141-151.
[BP2] Bedford, E. and Pinchuk, S., Domains in ${{\mathbb C}}^{n+1}$ with non-compact automorphism groups, J. Geom. Anal. 1(1991), 165-191. MR 92f:32024
[BP3] Bedford, E. and Pinchuk, S., Convex domains with non-compact automorphism group (translated from Russian), Russian Acad. Sci. Sb. Math. 82(1995), 1-20.
[BP4] Bedford, E. and Pinchuk, S., Domains in ${{\mathbb C}}^2$ with non-compact automorphism group, Indiana Univ. Math. J. 47(1998), 199-222. MR 99e:32058
[Ber1] Berteloot, F., Sur certains domaines faiblement pseudoconvexes dont le groupe d'automorphismes analytiques est non compact, Bull. Sci. Math. (2) 114(1990), 411-420. MR 91j:32035
[Ber2] Berteloot, F., Un principe de localisation pour les domaines faiblement pseudoconvexes de ${{\mathbb C}}^2$ dont le groupe d'automorphismes holomorphes est non compact, Colloque d'Analyse Complexe et Géométrie (Marseille, 1992), Asterisque 217(1993), 13-27. MR 94k:32019
[Ber3] Berteloot, F., Characterization of models in ${{\mathbb C}}^2$ by their automorphism groups, Internat. J. Math. 5(1994), 619-634. MR 95i:32040
[BeCo] Berteloot, F. and C $\oe$ uré G., Domaines de ${{\mathbb C}}^2$ , pseudoconvex et de type fini ayant un groupe non compact d'automorphismes, Ann. Inst. Fourier Grenoble 41(1991), 77-88. MR 92j:32053
[BM] Bochner, S. and Martin, W., Several complex variables, Princeton Univ. Press, 1948. MR 10:366a
[FIK1] Fu, S., Isaev, A. V. and Krantz, S. G., Examples of domains with non-compact automorphism groups, Math. Res. Letters 3(1996), 609-617. MR 97f:32047
[FIK2] Fu, S., Isaev, A. V. and Krantz, S. G., Reinhardt domains with non-compact automorphism groups, Math. Res. Letters 3(1996), 109-122. MR 97c:32043
[Fr] Frankel, S., Complex geometry of convex domains that cover varieties, Acta Math. 163(1989), 109-149. MR 90i:32037
[GK1] R. E. Greene and S. G. Krantz, Characterizations of certain weakly pseudo-convex domains with non-compact automorphism groups, in Complex Analysis Seminar, Springer Lecture Notes Vol. 1268 (1987), 121-157. MR 89c:32044
[IK1] A. Isaev and S. G. Krantz, Domains with non-compact automorphism group: A survey, Advances in Math. 146(1999), 1-38. MR 2000i:32053
[IK2] Isaev, A. V. and Krantz, S. G., On the boundary orbit accumulation set for a domain with non-compact automorphism group, Michigan Math. J. 43(1996), 611-617. MR 97m:32048
[IK3] Isaev, A. V. and Krantz, S. G., Finitely smooth Reinhardt domains with non-compact automorphism group, Illinois. J. Math. 41(1997), 412-420. MR 98i:32054
[IK4] Isaev, A. V. and Krantz, S. G., Hyperbolic Reinhardt domains with non-compact automorphism group, Pacific J. Math. 184(1998), 149-160. MR 99h:32003
[Ki1] Kim, K.-T., Domains with non-compact automorphism groups, Recent Developments in Geometry (Los Angeles, CA, 1987), 249-262, Contemp. Math. 101, Amer. Math. Soc.,1989. MR 90m:32036
[Ki2] Kim, K.-T., Complete localization of domains with non-compact automorphism groups, Trans. Amer. Math. Soc. 319(1990), 139-153. MR 90i:32035
[Ki3] Kim, K.-T., Domains in ${{\mathbb C}}^n$ with a piecewise Levi flat boundary which possess a non-compact automorphism group, Math. Ann. 292(1992), 575-586. MR 93h:32024
[Ki4] Kim, K.-T., Geometry of bounded domains and the scaling techniques in several complex variables, Lecture Notes Series 13, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 96c:32036
[Ki5] Kim, K.-T., On a boundary point repelling automorphism orbits, J. Math. Anal. Appl. 179(1993), 463-482. MR 94m:32048
[Ki6] Kim, K.-T., Two examples for scaling methods in several complex variables, RIM-GARC Preprint Series, Seoul National University 95-53(1995).
[KIKR] Kim, K.-T. and Krantz, S. G., A crash course in the function theory of several complex variables, Complex geometric analysis in Pohang(1997), 3-37, Contemp. Math. 222, American Math. Society, Providence, RI, 1999. MR 99g:32002
[KRA1] Krantz, S. G., Function Theory of Several Complex Variables, $2^{\rm nd}$ Ed., Wadsworth, Belmont, 1992. MR 93c:32001
[KRA2] Krantz, S. G., Convexity in complex analysis, Proc. Symp. Pure Math., vol. 52 (E. Bedford, J. D'Angelo, R. Greene, and S. Krantz eds.), American Mathematical Society, Providence, R.I., 1991. MR 92g:32001
[LEM1] Lempert, L., La metrique de Kobayashi et la representation des domains sur la boule, Bull. Soc. Math. France 109(1981), 427-474. MR 84d:32036
[LEM2] Lempert, L., The Dolbeault complex in infinite dimensions, J. AMS 11(1998), 485-520. MR 99f:58007
[Mu] Mujica, J., Complex analysis in Banach spaces, North-Holland, 1986. MR 88d:46084
[NAR] Narasimhan, R., Several Complex Variables, University of Chicago Press, Chicago, 1971. MR 49:7470
[ROS] Rosay, J.-P., Sur une characterization de la boule parmi les domains de ${\mathbb C}^n$ par son groupe d'automorphismes, Ann. Inst. Four. Grenoble XXIX(1979), 91-97.
[WON] Wong, B., Characterization of the ball in ${\mathbb C}^n$ by its automorphism group, Invent. Math. 41(1977), 253-257.