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

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} Δu_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8πδ_{p_{1}} Δu_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8πδ_{p_{2}} \end{array} \text{ in }\quad Ω\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.

33 pages, to appear on Communications in Mathematical Physics, 2016