The complex Monge-Ampere equation and an application to uniformisation of surfaces

Abstract: We prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature is biholomorphic to $\mathbb{C}<sup>2$. This result confirms a special case of Yau's conjecture that a complete noncompact K\"ahler $n$-manifold with positive holomorphic bisectional curvature is biholomorphic to $\mathbb{C}<sup>n$. In contrast to all known results on Yau's conjecture, we do not need additional assumptions on the global/asymptotic geometry of the K\"ahler surface apart from completeness. Towards this end, we prove that the integral of the square of the Ricci form of a complete K\"ahler surface with positive sectional curvature is finite. The work of Chen and Zhu shows that this latter result implies that the surface is biholomorphic to $\mathbb{C}<sup>2$. The main new idea is the construction of a Lipschitz continuous plurisubharmonic weight function with finite Monge-Amp`ere mass. This weight function is obtained by solving a complex Monge-Amp`ere equation.