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Journal of the American Mathematical Society

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

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Invariant metrics on negatively pinched complete Kähler manifolds
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by Damin Wu and Shing-Tung Yau HTML | PDF
J. Amer. Math. Soc. 33 (2020), 103-133

Abstract:

We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Berg- man metric, if the complete Kähler manifold is simply-connected, with the sectional curvature bounded between two negative constants. In particular, we confirm two conjectures of R. E. Greene and H. Wu posted in 1979.
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Additional Information
  • Damin Wu
  • Affiliation: Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009 Storrs, Connecticut 06269-1009
  • MR Author ID: 799841
  • Email: damin.wu@uconn.edu
  • Shing-Tung Yau
  • Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
  • MR Author ID: 185480
  • ORCID: 0000-0003-3394-2187
  • Email: yau@math.harvard.edu
  • Received by editor(s): December 5, 2017
  • Received by editor(s) in revised form: June 19, 2019
  • Published electronically: October 7, 2019
  • Additional Notes: The first author was partially supported by the NSF grant DMS-1611745
    The second author was partially supported by the NSF grants DMS-1308244 and DMS-1607871
  • © Copyright 2019 by the authors
  • Journal: J. Amer. Math. Soc. 33 (2020), 103-133
  • MSC (2010): Primary 32Q05, 32Q15, 32Q20, 32Q45; Secondary 32A25
  • DOI: https://doi.org/10.1090/jams/933
  • MathSciNet review: 4066473