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

TL;DR

This paper investigates the solvability of a curvature equation with multiple singular sources on a torus, revealing a deep connection with Painlevé VI equations and establishing criteria for solution existence based on the singular point location.

Contribution

It establishes a novel link between curvature equations with singular sources on tori and Painlevé VI equations, providing a sharp criterion for solution existence.

Findings

Existence of solutions depends on the position of the singular point p.

A sharp criterion for solution existence is derived using Painlevé VI equations.

The study addresses a critical case where traditional apriori estimates fail.

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.