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arXiv · 2507.16230

Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations

Zhijie Chen·Ting-Jung Kuo·Chang-Shou Lin
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Abstract

We study the curvature equation with multiple singular sources on a torus \[\Delta u+e^{u}=8\pi \sum_{k=0}^{3}n_{k}\delta_{\frac{\omega_{k}}{2}}% +4\pi \left( \delta_{p}+\delta_{-p}\right) \quad \text{ on }\;E_{\tau}:=\mathbb{C}/(\mathbb Z+\mathbb{Z}\tau),\] where $n_k\in\mathbb N$ and $\delta_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlev\'{e} VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlev\'{e} VI equations.

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Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin. 2025-07-22. Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations. https://arxiv.org/abs/2507.16230

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