A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature
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.
positive holomorphic bisectional curvature
scientific article; zbMATH DE number 1320420
scientific article