Invariant metrics on negatively pinched complete Kähler manifolds

Wu, Damin; Yau, Shing-Tung

doi:10.1090/jams/933

Invariant metrics on negatively pinched complete Kähler manifolds
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by Damin Wu and Shing-Tung Yau

J. Amer. Math. Soc. 33 (2020), 103-133

DOI: https://doi.org/10.1090/jams/933

Published electronically: October 7, 2019

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Previous version: Original version posted October 7, 2019
Corrected version: Current version corrects publisher-introduced error in equation numbering within the paper’s appendix

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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