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

TL;DR

The paper addresses the uniformisation problem for complete noncompact Kähler surfaces with positive sectional curvature, proving they are biholomorphic to $\mathbb{C}^2$ under the stronger positive and bounded curvature assumption. The authors introduce a novel uniformly Lipschitz plurisubharmonic weight $\phi$ with finite Monge–Ampère mass by solving a complex Monge–Ampère equation on an exhaustion, which enables a finiteness result for $\int_X \mathrm{Rc}_\omega^2$. Through a heat-flow smoothing, Hörmander L^2 estimates for holomorphic sections, and Bedford–Taylor calculus, they deduce $\int_X \mathrm{Rc}_\omega^2 < \infty$ and then apply Chen–Zhu’s framework to conclude that the manifold is biholomorphic to $\mathbb{C}^2$. This work advances the Yau uniformisation program for noncompact Kähler manifolds and provides a robust analytic tool—the finite Monge–Ampère mass weight—for handling positive curvature in noncompact settings.

Abstract

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.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:weight']}
• Lemma 6
• proof