On the existence of parabolic actions in convex domains of \(\mathbb C^{k+1}\). (English) Zbl 1363.32016

Berteloot, François; Thu, Ninh Van

Czech. Math. J. 65, No. 3, 579-585 (2015).

Summary: We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric.

MSC:

32M05	Complex Lie groups, group actions on complex spaces
32H02	Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables

Keywords:

convex domains; automorphism group

References:

[1]	E. Bedford, S. Pinchuk: Domains in \(\mathbb{C}\)2 with noncompact automorphism groups. Indiana Univ. Math. J. 47 (1998), 199–222. · Zbl 0907.32012 · doi:10.1512/iumj.1998.47.1552
[2]	E. Bedford, S. Pinchuk: Domains in \(\mathbb{C}\) n+1 with noncompact automorphism group. J. Geom. Anal. 1 (1991), 165–191. · Zbl 0733.32014 · doi:10.1007/BF02921302
[3]	E. Bedford, S. I. Pinchuk: Domains in \(\mathbb{C}\)2 with noncompact groups of holomorphic automorphisms. Math. USSR, Sb. 63 (1989), 141–151; translation from Mat. Sb., Nov. Ser. 135 (177) (1988), 147–157, 271. (In Russian.) · Zbl 0668.32029 · doi:10.1070/SM1989v063n01ABEH003264
[4]	F. Berteloot: Principe de Bloch et estimations de la métrique de Kobayashi des domaines de \(\mathbb{C}\)2. J. Geom. Anal. 13 (2003), 29–37. (In French.) · Zbl 1040.32011 · doi:10.1007/BF02930994
[5]	F. Berteloot: Characterization of models in \(\mathbb{C}\)2 by their automorphism groups. Int. J. Math. 5 (1994), 619–634. · Zbl 0817.32010 · doi:10.1142/S0129167X94000322
[6]	F. Berteloot, G. Coeuré: Domaines de \(\mathbb{C}\)2, pseudoconvexes et de type fini ayant un groupe non compact d’automorphismes. Ann. Inst. Fourier 41 (1991), 77–86. (In French.) · Zbl 0711.32016 · doi:10.5802/aif.1249
[7]	J. Byun, H. Gaussier: On the compactness of the automorphism group of a domain. C. R., Math., Acad. Sci. Paris 341 (2005), 545–548. · Zbl 1086.32020 · doi:10.1016/j.crma.2005.09.018
[8]	R. E. Greene, S. G. Krantz: Techniques for studying automorphisms of weakly pseudoconvex domains. Several Complex Variables: Proceedings of the Mittag-Leffler Institute, Stockholm, Sweden, 1987/1988 (J. E. Fornæss, ed.). Math. Notes 38, Princeton University Press, Princeton, 1993, pp. 389–410. · Zbl 0779.32017
[9]	A. V. Isaev, S. G. Krantz: Domains with non-compact automorphism group: a survey. Adv. Math. 146 (1999), 1–38. · Zbl 1040.32019 · doi:10.1006/aima.1998.1821
[10]	H. Kang: Holomorphic automorphisms of certain class of domains of infinite type. Tohoku Math. J. (2) 46 (1994), 435–442. · Zbl 0817.32011 · doi:10.2748/tmj/1178225723
[11]	K.-T. Kim: On a boundary point repelling automorphism orbits. J. Math. Anal. Appl. 179 (1993), 463–482. · Zbl 0816.32015 · doi:10.1006/jmaa.1993.1362
[12]	K.-T. Kim, S. G. Krantz: Some new results on domains in complex space with noncompact automorphism group. J. Math. Anal. Appl. 281 (2003), 417–424. · Zbl 1035.32019 · doi:10.1016/S0022-247X(03)00003-9
[13]	K.-T. Kim, S. G. Krantz: Complex scaling and domains with non-compact automorphism group. Ill. J. Math. 45 (2001), 1273–1299. · Zbl 1065.32014
[14]	M. Landucci: The automorphism group of domains with boundary points of infinite type. Ill. J. Math. 48 (2004), 875–885. · Zbl 1065.32016
[15]	J.-P. Rosay: Sur une caractérisation de la boule parmi les domaines de \(\mathbb{C}\)n par son groupe d’automorphismes. Ann. Inst. Fourier 29 (1979), 91–97. (In French.) · Zbl 0402.32001 · doi:10.5802/aif.768
[16]	B. Wong: Characterization of the unit ball in \(\mathbb{C}\)n by its automorphism group. Invent. Math. 41 (1977), 253–257. · Zbl 0385.32016 · doi:10.1007/BF01403050