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 non-compact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau  on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in , which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.
"A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature." J. Differential Geom. 67 (3) 519 - 570, July 2004. https://doi.org/10.4310/jdg/1102091357