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arXiv · 2406.14007

On canonical metrics of complex surfaces with split tangent and related geometric PDEs

Hao Fang·Joshua Jordan
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Abstract

In this paper, we study bi-Hermitian metrics on complex surfaces with split holomorphic tangent bundle and construct 2 types of metric cones. We introduce a new type of fully non-linear geometric PDE on such surfaces and establish smooth solutions. As a geometric application, we solve the prescribed Bismut Ricci problem. In various settings, we obtain canonical metrics on 2 important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type $\mathcal{S}_{M}$.

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Hao Fang, Joshua Jordan. 2024-06-20. On canonical metrics of complex surfaces with split tangent and related geometric PDEs. https://doi.org/10.1515/crelle-2025-0021

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