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July 2004 A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature
Bing-Long Chen, Siu-Hung Tang, Xi-Ping Zhu
J. Differential Geom. 67(3): 519-570 (July 2004). DOI: 10.4310/jdg/1102091357

Abstract

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

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Bing-Long Chen. Siu-Hung Tang. Xi-Ping Zhu. "A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature." J. Differential Geom. 67 (3) 519 - 570, July 2004. https://doi.org/10.4310/jdg/1102091357

Information

Published: July 2004
First available in Project Euclid: 3 December 2004

zbMATH: 1100.32009
MathSciNet: MR2153028
Digital Object Identifier: 10.4310/jdg/1102091357

Rights: Copyright © 2004 Lehigh University

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Vol.67 • No. 3 • July 2004
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