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arXiv · 2511.06849

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

Ved Datar·Vamsi Pritham Pingali·Harish Seshadri
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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.

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Ved Datar, Vamsi Pritham Pingali, Harish Seshadri. 2025-11-10. The complex Monge-Ampere equation and an application to uniformisation of surfaces. https://arxiv.org/abs/2511.06849

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