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

We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature is biholomorphic to ${\mathbb{C}}^2$. This result confirms a special case of Yau’s conjecture that a complete noncompact Kähler $n$-manifold with positive holomorphic bisectional curvature is biholomorphic to ${\mathbb{C}}^n$. In contrast to all known results on Yau’s conjecture, we do not need additional assumptions on the global/asymptotic geometry of the Kähler surface apart from completeness. Towards this end, we prove that the integral of the square of the Ricci form of a complete Kähler 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 ${\mathbb{C}}^2$ . The main new idea is the construction of a Lipschitz continuous plurisubharmonic weight function with finite Monge-Ampère mass. This weight function is obtained by solving a complex Monge-Ampère equation.