A Uniformization Theorem Of Complete Noncompact K\"{a}hler Surfaces With Positive Bisectional Curvature

By Binglong Chen, Siu-Hung Tang, Xiping Zhu. 2002

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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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3a4d3ca7b60e7ea1942f2f1569853398d3b7a04d

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Recent Progress on Ricci Solitons (2009, background): “(Chen-Tang-Zhu [28] and Cao [12] for n = 2, Ni [78] for n ≥ 3) Let gαβ̄(t) be a complete non-flat ancient solution to the Kähler-Ricci flow with bounded and nonnegative holomorphic bisectional curvature on a noncompact complex manifold X.”
Hamilton-Perelman's Proof of the Poincar\'e Conjecture and the Geometrization Conjecture (2006, background): “Section 22 of [65], see also [30] for application to the Kähler-Ricci flow and uniformization problem in complex dimension two).”
A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow (2006, methodology): “Moreover in the Kähler case, as shown by Chen, Tang and the second author in [29] (for complex two dimension) and by Ni in [98] (for all dimensions), the condition of nonnegative curvature operator can be replaced by the weaker condition of nonnegative bisectional curvature.”
Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature (2005, methodology): “We also establish a uniqueness theorem for the Ricci flow on complete noncompact manifolds.”
Recent developments on the Hamilton’s Ricci Flow (2007, methodology): “The following partial affirmative answer, due to Chen-Tang-Zhu [20] in complex dimension n=2 and Chau-Tam [16] for all dimensions, was obtained via the Ricci flow.”