Invariant metrics on negatively pinched complete K\"ahler manifolds

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 m...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2020-01, Vol.33 (1), p.103
Main Authors: Damin Wu, Shing-Tung Yau
Format: Article
Language:English
Online Access:Get full text
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Summary: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.
ISSN:0894-0347
1088-6834
DOI:10.1090/jams/933