The domain geometry and the bubbling phenomenon of rank two Gauge theory

TL;DR

This paper investigates the relationship between the critical points of Green's function on a flat torus and the bubbling solutions of a specific PDE modeling a Chern-Simons gauge theory, revealing that exactly three critical points occur under certain conditions.

Contribution

It proves that the Green's function has exactly three critical points when fully bubbling solutions of Liouville type exist for the gauge theory PDE.

Findings

Green's function has exactly three critical points under bubbling conditions.

Necessary conditions for multiple bubble solutions are established.

The results link geometric properties of the domain to solution behavior.

Abstract

Let $\Omega$ be a flat torus and $G$ be the green's function of $-\Delta$ on $\Omega$. One intriguing mystery of $G$ is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for the following equation that describes a Chern-Simons model in Gauge theory: \begin{equation}\label{e103} \left\{ \begin{array}{ll} \Delta u_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8\pi\delta_{p_{1}} \Delta u_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8\pi\delta_{p_{2}} \end{array} \text{ in }\quad \Omega\right., \quad p_1-p_2 \mbox{ is a half period}, \end{equation} if fully bubbling solutions of Liouville type exist, $G$ has exactly three critical points. In addition we establish necessary conditions for the existence of fully bubbling solutions with multiple bubbles.