Sharp results for spherical metric on flat tori with conical angle 6$\pi$ at two symmetric points

TL;DR

This paper establishes criteria for the existence of solutions to a curvature equation on flat tori with two symmetric conical singularities, providing explicit solutions for specific torus geometries and exploring the structure of non-even solution families.

Contribution

It introduces the first analysis of non-even solution families for curvature equations with multiple singular sources on flat tori, including explicit solutions for rectangle and rhombus tori.

Findings

Criteria for existence of solutions with two singularities

Explicit solutions for rectangle and rhombus tori

Analysis of solution structure for all τ in the upper half-plane

Abstract

In this paper, we investigate the following curvature equation: \begin{equation} \Delta u+e^{u}=8\pi (\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau }\text{, }\tau \in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{\tau }$ represents a flat torus and $\frac{\omega _{k}}{2}$ is one of the half periods of $E_{\tau }$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} \Delta u+e^{u}=8\pi \delta _{0}\text{ in }E_{\tau }\text{, }\tau \in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $\tau $. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} \Delta u+e^{u}=8\pi n(\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau },\text{ }n\in \mathbb{N}\text{.} \end{equation*} As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.