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Invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group. (English) Zbl 1360.32008
The authors study the following unbounded non-hyperbolic strongly pseudoconvex domain \[ D_{n,m}:=\Big\{(z,w)\in \mathbb{C}^n\times \mathbb{C}^m:\;\|w\|^2\leq e^{-\|z\|^2}\Big\}. \]
First, the authors investigate the holomorphic sectional curvature and the Ricci curvature of the Bergman metric on \(D_{1,1}\). Then, the authors show that the Bergman and Kähler-Einstein metrics are equivalent on \(D_{n,m}\). In the appendix, Carathéodory and Kobayashi pseudo-metrics and pseudo-distances are considered.

MSC:
32F45 Invariant metrics and pseudodistances in several complex variables
32T15 Strongly pseudoconvex domains
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[1] Azukawa, K; Suzuki, M, The Bergman metric on a thullen domain, Nagoya Math. J., 89, 1-11, (1983) · Zbl 0516.32009
[2] Bedford, E; Dadok, J, Bounded domains with prescribed group of automorphisms, Comment. Math. Helvetici, 62, 561-572, (1987) · Zbl 0647.32027
[3] Bedford, E; Pinchuk, S, Domains in \(\mathbb{C}^{n+l}\) with non-compact automorphism group, J. Geom. Anal., 1, 165-191, (1991) · Zbl 0733.32014
[4] Bergman, S.: The kernel function and conformal mapping, Mathematical Surveys, No. V. American Mathematical Society, Providence (1970)
[5] Bland, JS, The Einstein-kahler metric on \(\{\vert{ z}\vert ^2+\vert w\vert ^{2p}1\}\), Mich. Math. J., 33, 209-220, (1986) · Zbl 0602.32005
[6] Cheng, S-Y; Yau, S-T, On the existence of a complete kahler metric on noncompact complex manifolds and the regularity of fefferman’s equation, Commun. Pure Appl. Math., 33, 507-544, (1980) · Zbl 0506.53031
[7] Efimov, A.M.: A generalization of the Wong-Rosay theorem for the unbounded case. (Russian. Russian summary) Mat. Sb. 186, no. 7, pp. 41-50 (1995) (translation in Sb. Math. 186 (1995), no. 7, 967-976) · Zbl 0647.32027
[8] Engliš, M, Berezin quantization and reproducing kernels on complex domains, Trans. Am. Math. Soc., 348, 411-479, (1996) · Zbl 0842.46053
[9] Fu, S; Wong, B, On strictly pseudoconvex domains with kahler-Einstein Bergman metrics, Math. Res. Lett., 4, 697-703, (1997) · Zbl 0895.32005
[10] Grauert, H, Über modifikationen und exzeptionelle analytische mengen, Math. Ann., 146, 331-368, (1962) · Zbl 0173.33004
[11] Greene, R.E., Kim, K.-T., Krantz, S.G.: The geometry of complex domains. Progress in Mathematics, vol. 291, xiv, pp. 303. Birkh user Boston Inc, Boston (2011) · Zbl 1239.32011
[12] Harz, T; Shcherbina, N; Tomassini, G, Wermer type sets and extension of CR functions, Indiana Univ. Math. J., 61, 431-459, (2012) · Zbl 1271.32012
[13] Harz, T., Shcherbina, N., Tomassini, G.: On defining functions for unbounded pseudoconvex domains. arXiv:1405.2250v3 [math.CV] · Zbl 1386.32030
[14] Hirzebruch, F, Eulerian polynomials, Munster. J. Math., 1, 9-14, (2008) · Zbl 1255.11004
[15] Isaev, AV; Krantz, SG, Domains with noncompact automorphism group: a survey, Adv. Math., 146, 1-38, (1999) · Zbl 1040.32019
[16] Kim, K-T; Krantz, SG, Complex scaling and domains with non-compact automorphism group, Ill. J. Math., 45, 1273-1299, (2001) · Zbl 1065.32014
[17] Kim, H; Ninh, VT; Yamamori, A, The automorphism group of a certain unbounded non-hyperbolic domain, J. Math. Anal. Appl., 409, 637-642, (2014) · Zbl 1307.32017
[18] Klembeck, PF, Kähler metrics 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
[19] Kobayashi, S, On complete Bergman metrics, Proc. Am. Math. Soc., 13, 511-513, (1962) · Zbl 0114.04001
[20] Kosiński, Ł, Proper holomorphic mappings between Reinhardt domains in \(\mathbb{C}^2\), Mich. Math. J., 58, 711-721, (2009) · Zbl 1185.32008
[21] Ligocka, E, Forelli-rudin constructions and weighted Bergman projections, Stud. Math., 94, 257-272, (1989) · Zbl 0688.32020
[22] Mok, N., Yau, S.-T: Completeness of the Kähler-Einstein metric on bounded domain and the characterization of domain of holomorphy by curvature conditions. In: The mathematical heritage of Henri Poincaré, Part 1 (Bloomington, Ind., 1980), Proc. Sympos. Pure Math, vol. 39, pp. 41-49. American Mathematical Society, Providence (1983) · Zbl 0602.32005
[23] Nemirovskiĭ, S., Shafikov, R.G.: Conjectures of Cheng and Ramadanov. (Russian) Uspekhi Mat. Nauk 61(4) (370), 193-194 (2006) · Zbl 0647.32027
[24] Pflug, P; Zwonek, W, Bergman completeness of unbounded Hartogs domains, Nagoya Math. J., 180, 121-133, (2005) · Zbl 1094.32004
[25] Roos, G, Weighted Bergman kernels and virtual Bergman kernels, Sci. China Ser. A, 48, 225-237, (2005) · Zbl 1125.32001
[26] Rosay, J.-P.: Sur une caracterisation de la boule parmi les domaines de \(\mathbb{C}^n\) par son groupe dautomorphismes (French). Ann. Inst. Fourier (Grenoble) 29, no. 4, ix, 91-97 (1979) · Zbl 0402.32001
[27] Saerens, R; Zame, W, The isometry groups of manifolds and the automorphism groups of domains, Trans. Am. Math. Soc., 301, 413-429, (1987) · Zbl 0621.32025
[28] Springer, G, Pseudo-conformal transformations onto circular domains, Duke Math. J., 18, 411-424, (1951) · Zbl 0043.30401
[29] Tu, Z; Wang, L, Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains, J. Math. Anal. Appl., 419, 703-714, (2014) · Zbl 1293.32002
[30] Wang, A; Yin, W; Zhang, L; Roos, G, The kahler-Einstein metric for some Hartogs domains over symmetric domains, Sci. China Ser. A, 49, 1175-1210, (2006) · Zbl 1114.32011
[31] Wong, B, Characterization of the unit ball in \(\mathbb{C}^n\) by its automorphism group, Invent. Math., 41, 253-257, (1977) · Zbl 0385.32016
[32] Xu, ZY; Pan, GS, The geometric properties on a kind of Reinhardt domain (Chinese. English summary), J. Math. Study, 35, 49-55, (2002) · Zbl 1005.32003
[33] Yamamori, A, The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function, Complex Var. Elliptic Equ., 58, 783-793, (2013) · Zbl 1272.32002
[34] Yau, S-T; Yau, S-T (ed.), Problem section, seminar on differential geometry, No. 102, 669-706, (1982), Princeton
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