Journal of Differential Geometry

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Bing-Long Chen, Siu-Hung Tang, and Xi-Ping Zhu

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.

Article information

Source
J. Differential Geom., Volume 67, Number 3 (2004), 519-570.

Dates
First available in Project Euclid: 3 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1102091357

Digital Object Identifier
doi:10.4310/jdg/1102091357

Mathematical Reviews number (MathSciNet)
MR2153028

Zentralblatt MATH identifier
1100.32009

Citation

Chen, Bing-Long; Tang, Siu-Hung; Zhu, Xi-Ping. A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature. J. Differential Geom. 67 (2004), no. 3, 519--570. doi:10.4310/jdg/1102091357. https://projecteuclid.org/euclid.jdg/1102091357


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