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

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 noncompact complex two dimensional Kähler 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 Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

Volume Growth and Curvature Decay of Positively Curved Kahler Manifolds

“…On the other hand, Yau predicted in [26] that if the volume growth is of Euclidean, then the curvature must decay quadratically in certain average sense. In [8], under an additional assumption that the scalar curvature tends to zero at infinity in the average sense, Tang and the authors confirmed this for the complex surface case by using some special features in dimension 2 such as the Gauss-Bonnet-Chern formula and the classification of holonomy algebras for 4-dimensional Riemannian manifolds. The third result of the present paper is the following affirmative answer to this prediction of Yau for all dimensions under a more restricted curvature assumption.…”

“…The well-known conjecture of Yau on uniformization theorems asks if a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to a complex Euclidean space. The combination of the above Theorem 2 and the main theorem of [8] gives the following partial affirmative answer to the Yau's conjecture. 4 , for all 0 ≤ r < +∞, for some point x 0 ∈ M and some positive constant c. Then M is biholomorphic to the complex Euclidean space C 2 .…”

“…In [8] we established this result for complex 2-dimensional case under an additional assumption that the initial curvature decays to zero at infinity, where we used some special features of complex 2-dimension such as the classification of holonomy algebras and the splitting results in the (real)4-dimensional Riemannian manifolds obtained by the combination of Berger [1] and Hamilton [11].…”

“…After obtain the time decay estimate, we can adapt the argument in [8] to obtain the space decay estimates stated in Theorem 3. For sake of completeness, we present a proof as follows.…”

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

“…Section 22 of [63], see also [29] for application to the Kähler-Ricci flow and uniformization problem in complex dimension two). A special consequence of the finite bump theorem is that if we have a complete noncompact solution to the Ricci flow on an ancient time interval −∞ < t < T with T > 0 satisfying certain local injectivity radius bound, with curvature bounded at each time and with asymptotic scalar curvature ratio A = lim sup Rs 2 = ∞, then we can find a sequence of points p j going to ∞ (as in the following Lemma 6.1.3) such that a cover of the limit of dilations around these points at time t = 0 splits as a product with a flat factor.…”