The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

Chen, Bo-Yong; Zhang, Jin-Hao

doi:10.1090/S0002-9947-02-02989-6

The Bergman metric on a Stein manifold with a bounded plurisubharmonic function
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by Bo-Yong Chen and Jin-Hao Zhang PDF

Trans. Amer. Math. Soc. 354 (2002), 2997-3009 Request permission

Abstract:

In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature $\le -A/\rho ^2$, which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.

References

Zbigniew Błocki and Peter Pflug, Hyperconvexity and Bergman completeness, Nagoya Math. J. 151 (1998), 221–225. MR 1650305, DOI 10.1017/S0027763000025265
B. Y. Chen, The Bergman metric on complete Kähler manifolds, preprint.
B. Y. Chen and J. H. Zhang, Bergman exhaustivity, completeness and stability, Adv. Math. (Chinese) 29 (2000), 397–410.
———-, On Bergman completeness and Bergman stability Math. Ann. 318 (2000), 517–526.
Klas Diederich and Takeo Ohsawa, An estimate for the Bergman distance on pseudoconvex domains, Ann. of Math. (2) 141 (1995), no. 1, 181–190. MR 1314035, DOI 10.2307/2118631
R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983
Gregor Herbort, The Bergman metric on hyperconvex domains, Math. Z. 232 (1999), no. 1, 183–196. MR 1714284, DOI 10.1007/PL00004754
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Jürgen Jost and Kang Zuo, Vanishing theorems for $L^2$-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry, Comm. Anal. Geom. 8 (2000), no. 1, 1–30. MR 1730897, DOI 10.4310/CAG.2000.v8.n1.a1
M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), no. 2, 231–240 (English, with French summary). MR 820321
Shoshichi Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290. MR 112162, DOI 10.1090/S0002-9947-1959-0112162-5
Shoshichi Kobayashi, On complete Bergman metrics, Proc. Amer. Math. Soc. 13 (1962), 511–513. MR 141795, DOI 10.1090/S0002-9939-1962-0141795-0
Takeo Ohsawa, Boundary behavior of the Bergman kernel function on pseudoconvex domains, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, 897–902. MR 764336, DOI 10.2977/prims/1195180870
Rolf Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257–286 (German). MR 222334, DOI 10.1007/BF02063212
Nessim Sibony, A class of hyperbolic manifolds, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 357–372. MR 627768
J. L. Walsh, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc. 46 (1939), 46–65. MR 55, DOI 10.1090/S0002-9947-1939-0000055-0
Jean-Luc Stehlé, Fonctions plurisousharmoniques et convexité holomorphe de certains fibrés analytiques, Séminaire Pierre Lelong (Analyse), Année 1973–1974, Lecture Notes in Math., Vol. 474, Springer, Berlin, 1975, pp. 155–179 (French). MR 0399524
Włodzimierz Zwonek, Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions, Dissertationes Math. (Rozprawy Mat.) 388 (2000), 103. MR 1785672, DOI 10.4064/dm388-0-1