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

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

DOI: https://doi.org/10.1090/S0002-9947-02-02989-6

Published electronically: March 29, 2002

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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.

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