A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature

Details A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature

2002-11-24

arxiv.org/abs/math/0211372v1

Mathematics

Differential Geometry

57 pages

Scientific paper

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 noncompact complex two dimensional K\"ahler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a K\"ahler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

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