Abstract
We consider certain pseudoconvex domains in Cn+1 and show that if the automorphism group is noncompact, then the domain is equivalent to\(E_m = \{ |w|^2 + |z_1 |^{2m} + |z_2 |^2 + \cdots + |z_n |^2< 1\} \) for some integerm ≥ 1.
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References
Bedford, E., and Fornæss, J.-E. A construction of peak functions on weakly pseudoconvex domains. Ann. Math.107, 555–568 (1978).
Bedford, E., and Pinchuk, S. Domains in C2 with noncompact group of automorphisms. (Russian) Mat. Sb.135, 147–157 (1988); (English) Math. USSR Sbornik63, 141–151 (1989).
Bell, S., and Catlin, D. Personal communication.
Berteloot, F., and Cœuré, G. Domaines de C2, pseudoconvexes et de type fini ayant un groupe non compact d’automorphismes. Preprint.
Catlin, D. Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z.200, 429–466 (1989).
Diederich, K., and Fornæss, J.-E. Pseudoconvex domains with real-analytic boundary. Ann. Math.107, 371–384 (1978).
Diederich, K., and Fornæss, J.-E. Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions. Invent. Math.39, 129–141 (1977).
Frankel, S. Complex geometry of convex domains that cover varieties. Acta. Math.163, 109–149 (1989).
Greene, R., and Krantz, S. Characterizations of certain weakly pseudoconvex domains with non-compact automorphism groups. In: Complex Analysis Seminar, edited by S. Krantz, Springer Lecture Notes, Vol. 1268, pp. 121–157. Springer-Verlag 1987.
Greene, R., and Krantz, S. Invariants of Bergman geometry and results concerning the automorphism groups of domains in Cn. Preprint.
Kim, K.-T. Domains with noncompact automorphism groups. Contemp. Math. AMS101, 249–262 (1989).
Kim, K.-T. Complete localization of domains with noncompact automorphism groups. Trans. AMS319, 130–153 (1990).
Klembeck, P. Kähler metrics of negative curvature, the Bergman metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex domains. Indiana U. Math. J.27, 275–282 (1978).
Kodama, A. On the structure of a bounded domain with a special boundary point. Osaka J. Math.23, 271–298 (1986).
Kodama, A. On the structure of a bounded domain with a special boundary point II. Osaka J. Math.24, 499–519 (1987).
Kodama, A. Characterizations of certain weakly pseudoconvex domainsE(κ, α) in Cn. Tôhoku Math. J.40, 343–365 (1988).
Kodama, A. A characterization of certain domains with good boundary points in the sense of Greene-Krantz. Kodai Math. J.12, 257–269 (1989).
Krantz, S. Convexity in complex analysis. In: Proc. Symp. Pure Math. Providence, RI: Amer. Math. Soc. To appear.
Narasimhan, R. Several Complex Variables. Chicago, IL: University of Chicago Press 1971.
Pinchuk, S. The scaling method in several complex variables. In: Proc. Symp. Pure Math. Providence, RI: Amer. Math. Soc.To appear.
Rosay, J.-P. Sur une caractérisation de la boule parmi les domaines de Cn par son groupe d’autmorphismes. Ann. Inst. Four. GrenobleXXIX, 91–97 (1979).
Sibony, N. A class of hyperbolic manifolds. In: Recent Developments in Several Complex Variables, edited by J.-E. Fornæss, Annals of Mathematics Studies 100. Princeton, NJ: Princeton University Press 1981.
Wong, B. Characterization of the ball in Cn by its automorphism group. Invent. Math.41, 253–257 (1977).
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Communicated by Steven Krantz
First author supported in part by a grant from the National Science Foundation. This research was carried out while the second author was visiting Indiana University.
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Bedford, E., Pinchuk, S. Domains in Cn+1 with noncompact automorphism group. J Geom Anal 1, 165–191 (1991). https://doi.org/10.1007/BF02921302
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DOI: https://doi.org/10.1007/BF02921302