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

Ting-Jung Kuo

Abstract

In this paper, we investigate the following curvature equation: \begin{equation} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ}\text{, }τ\in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{τ}$ represents a flat torus and $\frac{ω_{k}}{2}$ is one of the half periods of $E_{τ}$. 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*} Δu+e^{u}=8πδ_{0}\text{ in }E_{τ}\text{, }τ\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 $τ$. 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*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ},\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.

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

Abstract

In this paper, we investigate the following curvature equation: \begin{equation} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ}\text{, }τ\in \mathbb{H} (0.1) \label{a} \end{equation} Here represents a flat torus and is one of the half periods of . 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*} Δu+e^{u}=8πδ_{0}\text{ in }E_{τ}\text{, }τ\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 . 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*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})\text{ in } E_{τ},\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.
Paper Structure (8 sections, 44 theorems, 373 equations, 1 figure)

This paper contains 8 sections, 44 theorems, 373 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integrability of the curvature equation
  3. Normalized developing map
  4. Unitary monodromy constraints
  5. Spectral curve and Baker-Akhiezer function
  6. Monodromy theory for equation (\ref{['1999']})
  7. Non-even family of solutions and pre-modular forms
  8. The proof of Theorem \ref{['thm21 copy(1)']}

Key Result

Theorem 1.2

Equation (164) defined on $E_{\tau_{0}}$, where $\tau_{0}\in\mathbb{H}$, has a non-even family of solutions if and only if equation (1631), defined on the same $E_{\tau_{0}}$, has a solution, i.e., there is $(r,s)$$\in$$\mathbb{R}^{2}$$\setminus$$\frac{1}{2}\mathbb{Z}^{2}$ such that $Z(r,s,\tau_{0})

Figures (1)

  • Figure 1: Doamin $\Lambda$

Theorems & Definitions (78)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • ...and 68 more