A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature
type: Item
instance of: scholarly article
title: A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature
zbMATH Open document ID: 0931.53034
DOI: 10.1007/BF02922111
author: Wan-Xiong Shi
published in: The Journal of Geometric Analysis
publication date: 2000-02-20T00:00:00Z
review text: A famous theorem by Mori and by Siu and Yau states that any compact Kähler manifold of complex dimension \(n\) with positive holomorphic bisectional curvature is biholomorphic to \(\mathbb{C} P^n\). The corresponding conjecture for noncompact manifolds due to Greene and Wu is as follows. Conjecture. Any complete noncompact Kähler manifold of complex dimension \(n\) with positive holomorphic bisectional curvature is biholomorphic to \(\mathbb{C}^n\). In the present paper the following partial result is shown: Theorem. Let \(M\) be a complete noncompact Kähler manifold of complex dimension \(n\geq 2\) with bounded and positive holomorphic bisectional curvature. In addition, assume that there exists a constant \(C\) such that for all \(x_0\in M\) and for all \(r>0\) \[ \int_{B(x_0,r)} \text{scal}(x)dx \leq {C \over (r+1)^2} \text{Vol}(B(x_0,r)) \] holds where scal denotes scalar curvature and \(B(x_0,r)\) is the geodesic ball of radius \(r\) centered at \(x_0\). Then the universal covering space of \(M\) is biholomorphic to \(\mathbb{C}^n\). The proof uses Hamilton's Ricci flow \({\partial \over \partial t} g_{ij} = -2 \text{Ric}_{ij}\). It is shown that this evolution equation has a solution for all times \(0 \leq t < \infty\) and that the curvature of \(g_{ij}(t)\) tends to zero as \(t \to \infty\). This yields a flat Kähler metric on the universal covering space.
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1320420
zbMATH Keywords: complete Kähler manifolds
zbMATH Keywords: positive holomorphic bisectional curvature
zbMATH Keywords: Ricci flow
zbMATH Keywords: uniformization theorem
reviewed by: Christian Bär
MaRDI profile type: Publication
cites work: On the classification of three-dimensional compact Kähler manifolds of nonnegative bisectional curvature
cites work: Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes
cites work: Q5341449
cites work: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens
cites work: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds
cites work: Q4067705
cites work: On complete open manifolds of positive curvature
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Some isoperimetric inequalities and eigenvalue estimates
cites work: Deforming metrics in the direction of their Ricci tensors
cites work: Bounded harmonic functions and positive Ricci curvature
cites work: Q4163752
cites work: \(C^\infty\) convex functions and manifolds of positive curvature
cites work: Function theory on manifolds which possess a pole
cites work: On compact Kaehler manifolds of nonnegative bisectional curvature. I
cites work: Three-manifolds with positive Ricci curvature
cites work: Four-manifolds with positive curvature operator
cites work: Ricci deformation of the metric on a Riemannian manifold
cites work: On the parabolic kernel of the Schrödinger operator
cites work: Q3312496
cites work: An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties
cites work: The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature
cites work: On Harnack's theorem for elliptic differential equations
cites work: Q5533735
cites work: Deforming the metric on complete Riemannian manifolds
cites work: Ricci deformation of the metric on complete non-compact Riemannian manifolds
cites work: Complete noncompact three-manifolds with non-negative Ricci curvature
cites work: Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature
cites work: Q3937980
cites work: On the transitivity of holonomy systems
cites work: On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I
description: scientific article; zbMATH DE number 1320420