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

TL;DR

This paper proves that certain complete noncompact Kähler surfaces with positive sectional curvature are biholomorphic to ^2, confirming a special case of Yau's conjecture without additional geometric assumptions.

Contribution

It establishes a new result confirming Yau's conjecture for Kähler surfaces with positive sectional curvature, using a novel approach involving a Lipschitz continuous plurisubharmonic weight function.

Findings

Complete noncompact Kähler surface with positive sectional curvature is biholomorphic to ^2.

Finite integral of the Ricci form squared for such surfaces.

Construction of a Lipschitz continuous plurisubharmonic weight function via Monge-Ampère equation.

Abstract

We prove that a complete noncompact K\"ahler 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 K\"ahler $n$-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 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 $\mathbb{C}^2$ . 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.