The complex Monge-Ampere equation and an application to uniformisation of surfaces
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| Format: | Preprint |
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2025
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| _version_ | 1866911257749094400 |
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| author | Datar, Ved Pingali, Vamsi Pritham Seshadri, Harish |
| author_facet | Datar, Ved Pingali, Vamsi Pritham Seshadri, Harish |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The complex Monge-Ampere equation and an application to uniformisation of surfaces Datar, Ved Pingali, Vamsi Pritham Seshadri, Harish Differential Geometry 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. |
| title | The complex Monge-Ampere equation and an application to uniformisation of surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.06849 |