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Domains with non-compact automorphism group: a survey. (English) Zbl 1040.32019
Summary: We survey results arising from the study of domains in \(\mathbb C^n\) with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points. Particular attention will be paid to results derived in the past ten years.

MSC:
32M05 Complex Lie groups, group actions on complex spaces
32T15 Strongly pseudoconvex domains
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[1] Bedford, E.; Dadok, J., Bounded domains with prescribed group of automorphisms, Comment math. helv., 62, 561-572, (1987) · Zbl 0647.32027
[2] Bedford, E.; Pinchuk, S., Domains in \(C\)^{2} with non-compact holomorphic automorphism group, Math. USSR-sb., 63, 141-151, (1989) · Zbl 0668.32029
[3] Bedford, E.; Pinchuk, S., Domains in \(C\)^{n+1} with non-compact automorphism groups, J. geom. anal., 1, 165-191, (1991) · Zbl 0733.32014
[4] Bedford, E.; Pinchuk, S., Convex domains with non-compact automorphism group, Russian acad. sci. sb. math., 82, 1-20, (1995) · Zbl 0847.32023
[5] Bedford, E.; Pinchuk, S., Domains in \(C\)^{2} with non-compact automorphism group, Indiana univ. math. J., 47, 149-222, (1999)
[6] Bell, S., Biholomorphic mappings and the ∂ problem, Ann. of math., 114, 103-113, (1981) · Zbl 0423.32009
[7] Bell, S., Compactness of families of holomorphic mappings up to the boundary, Complex analysis, Lecture notes in mathematics, 1268, (1987), Springer-Verlag New York/Berlin, p. 29-42 · Zbl 0633.32020
[8] Bell, S., Weakly pseudoconvex domains with non-compact automorphism groups, Math. ann., 280, 403-408, (1988) · Zbl 0617.32030
[9] S. Bell, and, D. Catlin, personal communication.
[10] Berteloot, F.; Cœuré, G., Domaines de \(C\)^{2}, pseudoconvex et de type fini ayant un groupe non compact d’automorphismes, Ann. inst. Fourier (Grenoble), 41, 77-88, (1991) · Zbl 0711.32016
[11] Berteloot, F., Sur certains domaines faiblement pseudoconvexes dont le groupe d’automorphismes analytiques est non compact, Bull. sci. math. (2), 114, 411-420, (1990) · Zbl 0717.32018
[12] Berteloot, F., Un principe de localisation pour LES domaines faiblement pseudoconvexes de \(C\)^{2} dont le groupe d’automorphismes holomorphes est non compact, Asterisque, 217, 13-27, (1993) · Zbl 0794.32018
[13] Berteloot, F., Characterization of models in \(C\)^{2} by their automorphism groups, Internat. J. math., 5, 619-634, (1994) · Zbl 0817.32010
[14] Bland, J.; Duchamp, T.; Kalka, M., A characterization of \(C\)\(P\)^{n} by its automorphism group, Complex analysis, Lecture notes in mathematics, 1268, (1987), Springer-Verlag New York/Berlin, p. 60-65 · Zbl 0621.32030
[15] Braun, R.; Kaup, W.; Upmeier, H., On the automorphisms of circular and Reinhardt domains in complex Banach spaces, Manuscripta math., 25, 97-133, (1978) · Zbl 0398.32001
[16] Burns, D.; Shnider, S.; Wells, R.O., Deformations of strictly pseudoconvex domains, Invent. math., 46, 237-253, (1978) · Zbl 0412.32022
[17] Cartan, É., Sur LES domaines bornés, homogènes de l’espace de n variables complexes, Abh. math. sem. univ. Hamburg, 11, 116-162, (1936) · JFM 61.0370.03
[18] Catlin, D., Boundary invariants of pseudoconvex domains, Ann. of math., 120, 529-586, (1984) · Zbl 0583.32048
[19] Chern, S.S.; Moser, J., Real hypersurfaces in complex manifolds, Acta math., 133, 219-271, (1974) · Zbl 0302.32015
[20] Coupet, B.; Sukhov, A., On the boundary rigidity phenomenon for automorphisms of domains in \(C\)^{n}, Proc. amer. math. soc., 124, 3371-3380, (1996) · Zbl 0868.32018
[21] D’Angelo, J., Real hypersurfaces, orders of contact, and applications, Ann. of math., 115, 615-637, (1982) · Zbl 0488.32008
[22] D’Angelo, J., Several complex variables and the geometry of real hypersurfaces, (1993), CRC Press Boca Raton · Zbl 0854.32001
[23] Diederich, K.; Fornaess, J.E., Pseudoconvex domains with real analytic boundary, Ann. of math., 107, 371-384, (1978) · Zbl 0378.32014
[24] Efimov, A., Extension of the Wong-rosay theorem to the unbounded case, Sb. mat., 186, 967-976, (1995) · Zbl 0865.32020
[25] Fefferman, C., The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. math., 26, 1-65, (1974) · Zbl 0289.32012
[26] Frankel, S., Complex geometry of convex domains that cover varieties, Acta math., 163, 109-149, (1989) · Zbl 0697.32016
[27] Fu, S.; Isaev, A.V.; Krantz, S.G., Examples of domains with non-compact automorphism groups, Math. res. lett., 3, 609-617, (1996) · Zbl 0881.32001
[28] Fu, S.; Isaev, A.V.; Krantz, S.G., Reinhardt domains with non-compact automorphism groups, Math. res. lett., 3, 109-122, (1996) · Zbl 0866.32001
[29] Fu, S.; Wong, B., On boundary accumulation points of a smoothly bounded pseudoconvex domain in \(C\)^{2}, Math. ann., 310, 183-196, (1998) · Zbl 0955.32011
[30] S. Fu, and, B. Wong, On a domain in \(C\)^{2} with generic piecewise smooth Levi-flat boundary and non-compact automorphism group, preprint. · Zbl 1026.32047
[31] Gaussier, H., Characterization of convex domains with non-compact automorphism group, Michigan math. J., 44, 375-388, (1997) · Zbl 0889.32032
[32] Graham, I., Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \(C\)^{n} with smooth boundary, Trans. amer. math. soc., 207, 219-240, (1975) · Zbl 0305.32011
[33] Greene, R.E.; Krantz, S.G., Deformation of complex structures, estimates for the ∂ equation, and stability of the Bergman kernel, Adv. math., 43, 1-86, (1982) · Zbl 0504.32016
[34] Greene, R.E.; Krantz, S.G., Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings, Complex analysis of several complex variables, Proc. sympos. pure math., 41, (1984), Amer. Math. Soc Providence, p. 77-93
[35] Greene, R.E.; Krantz, S.G., Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, Indiana univ. math. J., 34, 865-879, (1985) · Zbl 0622.32020
[36] Greene, R.E.; Krantz, S.G., Characterization of certain weakly pseudoconvex domains with non-compact automorphism groups, Complex analysis, Lecture notes in mathematics, 1268, (1987), Springer-Verlag New York/Berlin, p. 121-157
[37] Greene, R.E.; Krantz, S.G., Biholomorphic self-maps of domains, Complex analysis II, Lecture notes in mathematics, 1276, (1987), Springer-Verlag New York/Berlin, p. 136-207 · Zbl 0625.32024
[38] Greene, R.E.; Krantz, S.G., Invariants of Bergman geometry and the automorphism groups of domains in \(C\)^{n}, Geometrical and algebraical aspects in several complex variables, Sem. conf. 8, (1991), EditEl Rende, p. 107-136 · Zbl 0997.32012
[39] Greene, R.E.; Krantz, S.G., Techniques for studying automorphisms of weakly pseudoconvex domains, Several complex variables, Math. notes, 38, (1993), Princeton Univ. Press Princeton, p. 389-410 · Zbl 0779.32017
[40] Huang, X., Some applications of Bell’s theorem to weakly pseudoconvex domains, Pacific J. math., 158, 305-315, (1993) · Zbl 0807.32016
[41] Huckleberry, A.; Oeljeklaus, K., Classification theorems for almost homogeneous spaces, Institut élie Cartan, (1984), Université de NancyInstitut Élie Cartan Nancy · Zbl 0549.32024
[42] Isaev, A.V.; Krantz, S.G., On the boundary orbit accumulation set for a domain with non-compact automorphism group, Michigan math. J., 43, 611-617, (1996) · Zbl 0879.32016
[43] Isaev, A.V.; Krantz, S.G., Finitely smooth Reinhardt domains with non-compact automorphism group, Illinois J. math., 41, 412-420, (1997) · Zbl 0879.32002
[44] Isaev, A.V.; Krantz, S.G., Hyperbolic Reinhardt domains with non-compact automorphism group, Pacific J. math., 184, 149-160, (1998) · Zbl 0918.32013
[45] Kim, K.-T., Domains with non-compact automorphism groups, Recent developments in geometry, Contemp. math., 101, (1989), Amer. Math. Soc Providence, p. 249-262
[46] Kim, K.-T., Complete localization of domains with non-compact automorphism groups, Trans. amer. math. soc., 319, 139-153, (1990) · Zbl 0705.32008
[47] Kim, K.-T., Domains in \(C\)^{n} with a piecewise Levi flat boundary which possess a non-compact automorphism group, Math. ann., 292, 575-586, (1992) · Zbl 0735.32021
[48] Kim, K.-T., Geometry of bounded domains and the scaling techniques in several complex variables, Lecture notes series, (1993), Seoul National UniversityResearch Institute of Mathematics, Global Analysis Research Center Seoul
[49] Kim, K.-T., On a boundary point repelling automorphism orbits, J. math. anal. appl., 179, 463-482, (1993) · Zbl 0816.32015
[50] K.-T. Kim, Two examples for scaling methods in several complex variables, RIM-GARC preprint Series, Seoul National University, 95-53, 1995.
[51] Kim, K.-T.; Yu, J., Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains, Pacific J. math., 176, 141-163, (1996) · Zbl 0886.32020
[52] Klembeck, P., Kähler metric of negative curvature, the bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana univ. math. J., 27, 275-282, (1978) · Zbl 0422.53032
[53] Kobayashi, S., Hyperbolic manifolds and holomorphic mappings, (1970), Dekker New York · Zbl 0207.37902
[54] Kobayashi, S., Intrinsic distances, measures and geometric function theory, Bull. amer. math. soc., 82, 357-416, (1976) · Zbl 0346.32031
[55] Kodama, A., A remark on bounded Reinhardt domains, Proc. Japan acad. ser. A math. sci., 54, 179-182, (1978) · Zbl 0412.32001
[56] Kodama, A., On the structure of a bounded domain with a special boundary point, Osaka J. math., 23, 271-298, (1986) · Zbl 0608.32012
[57] Kodama, A., On the structure of a bounded domain with a special boundary point, II, Osaka J. math., 24, 499-519, (1987) · Zbl 0629.32027
[58] Kodama, A., Characterization of certain weakly pseudoconvex domains E(k, α) in \(C\)n, Tôhoku math. J., 40, 343-365, (1988) · Zbl 0667.32013
[59] Kodama, A., A characterization of certain domains with good boundary points in the sense of Greene-krantz, Kodai. math. J., 12, 257-269, (1989) · Zbl 0693.32008
[60] Kodama, A., Characterization of certain weakly pseudoconvex domains in \(C\)^{n} from the viewpoint of biholomorphic automorphism groups, Several complex variables and complex geometry, part 2, Proc. sympos. pure math., 52, (1991), Amer. Math. Soc Providence, p. 291-296 · Zbl 0739.32018
[61] Kodama, A., A characterization of certain domains with good boundary points in the sense of Greene-krantz, II, Tôhoku math. J., 43, 9-25, (1991) · Zbl 0736.32004
[62] Kodama, A., A characterization of certain domains with good boundary points in the sense of Greene-krantz, III, Osaka J. math., 32, 1055-1063, (1995) · Zbl 0857.32008
[63] Kodama, A.; Krantz, S.G.; Ma, D.W., A characterization of generalized complex ellipsoids in \(C\)^{n} and related results, Indiana univ. math. J., 41, 173-195, (1992)
[64] Krantz, S.G., Characterization of smooth domains in \(C\) by their biholomorphic self-maps, Amer. math. monthly, 90, 555-557, (1983) · Zbl 0524.30007
[65] Krantz, S.G., Convexity in complex analysis, Several complex variables and complex geometry, part 1, Proc. sympos. pure math., 52, (1991), Amer. Math. Soc Providence, p. 119-137 · Zbl 0739.32019
[66] Krantz, S.G., Function theory of several complex variables, (1992), Wadsworth Belmont · Zbl 0776.32001
[67] S. G. Krantz, Survey of some recent ideas concerning automorphism groups of domains, in, Proceedings of a Conference in Honor of Pierre Dolbeault, Hermann, Paris, 1995. · Zbl 0963.32012
[68] Kruzhilin, N.G., Holomorphic automorphisms of hyperbolic Reinhardt domains, Math. USSR-izv., 32, 15-38, (1989) · Zbl 0663.32019
[69] Lempert, L., On the boundary behavior of holomorphic mappings, Contributions to several complex variables, Aspects of math., E9, (1986), Vieweg Braunschweig, p. 193-215
[70] Lempert, L.; Rubel, L., An independence result in several complex variables, Proc. amer. math. soc., 113, 1055-1065, (1991) · Zbl 0737.03027
[71] Morimoto, A.; Nagano, T., On pseudo-conformal deformations of hypersurfaces, J. math. soc. Japan, 15, 289-300, (1963) · Zbl 0119.06701
[72] Narasimhan, R., Several complex variables, (1971), Univ. of Chicago Press Chicago · Zbl 0223.32001
[73] Pinchuk, S., Holomorphic inequivalence of some classes of domains in \(C\)^{n}, Math. USSR-sb., 39, 61-86, (1981) · Zbl 0464.32014
[74] Pinchuk, S., Homogeneous domains with piecewise smooth boundaries, Math. notes, 32, 849-852, (1983) · Zbl 0576.32041
[75] Pinchuk, S., The scaling method and holomorphic mappings, Several complex variables and complex geometry, part 1, Proc. sympos. pure math., 52, (1991), Amer. Math. Soc Providence, p. 151-161 · Zbl 0744.32013
[76] Poincaré, H., LES fonctions analytiques de deux variables et la représentation conforme, Rend. circ. mat. Palermo, 23, 185-220, (1907) · JFM 38.0459.02
[77] Poletskii, E.A.; Shabat, B.V., Invariant metrics, Encycl. math. sci., (1989), Springer-Verlag New York/Berlin, p. 63-111
[78] Pyatetskii-Shapiro, I., Automorphic functions and the geometry of classical domains, (1969), Gordon & Breach New York
[79] Rosay, J.P., Sur une caractérisation de la boule parmi LES domaines de \(C\)^{n} par son groupe d’automorphismes, Ann. inst. Fourier (Grenoble), 29, 91-97, (1979) · Zbl 0402.32001
[80] Rudin, W., Function theory in the unit ball of \(C\)^{n}, (1980), Springer-Verlag New York/Berlin
[81] Saerens, R.; Zame, W., The isometry groups of manifolds and the automorphism groups of domains, Trans. amer. math. soc., 301, 413-429, (1987) · Zbl 0621.32025
[82] Shimizu, S., Automorphisms of bounded Reinhardt domains, Japan J. math., 15, 385-414, (1989) · Zbl 0712.32003
[83] Sunada, T., Holomorphic equivalence problem for bounded Reinhardt domains, Math. ann., 235, 111-128, (1978) · Zbl 0357.32001
[84] Tumanov, A., Geometry of CR-manifolds, Encycl. math. sci., (1989), Springer-Verlag New York/Berlin, p. 201-221 · Zbl 0658.32007
[85] Tumanov, A.; Shabat, G., Realization of linear Lie groups by biholomorphic automorphisms of bounded domains, Funct. anal. appl., 24, 255-257, (1991) · Zbl 0717.32019
[86] Wong, B., Characterization of the unit ball in \(C\)^{n} by its automorphism group, Invent. math., 41, 253-257, (1977) · Zbl 0385.32016
[87] Wong, B., Characterization of the bidisc by its automorphism group, Amer. J. math., 117, 279-288, (1995) · Zbl 0827.32022
[88] Zaitsev, D., On the automorphism group of algebraic bounded domains, Math. ann., 302, 105-129, (1995) · Zbl 0823.14005
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