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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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Additional Information
  • Bo-Yong Chen
  • Affiliation: Department of Applied Mathematics, Tongji University, Shanghai 200092, China
  • Email: chenboy@online.sh.cn
  • Jin-Hao Zhang
  • Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, China
  • Email: zhangjhk@online.sh.cn
  • Received by editor(s): August 1, 2001
  • Published electronically: March 29, 2002
  • Additional Notes: The first author was supported by an NSF grant TY10126005 and a grant from Tongji Univ. No. 1390104014
    The second author was supported by project G1998030600
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2997-3009
  • MSC (2000): Primary 32H10
  • DOI: https://doi.org/10.1090/S0002-9947-02-02989-6
  • MathSciNet review: 1897387