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Journal of the American Mathematical Society

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

   
 
 

 

Invariant metrics on negatively pinched complete Kähler manifolds


Authors: Damin Wu and Shing-Tung Yau
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 32Q05, 32Q15, 32Q20, 32Q45; Secondary 32A25
DOI: https://doi.org/10.1090/jams/933
Published electronically: October 7, 2019
Previous version: Original version posted October 7, 2019
Corrected version: Current version corrects publisher-introduced error in equation numbering within the paper’s appendix
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Abstract | References | Similar Articles | Additional Information

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
Email: damin.wu@uconn.edu

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: yau@math.harvard.edu

DOI: https://doi.org/10.1090/jams/933
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
Article copyright: © Copyright 2019 by the authors