Geometric quantities arising from bubbling analysis of mean field equations

Abstract

Let $E=\mathbb{C}/\Lambda$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$: $$G_n(z_1,\cdots ,z_n):=\sum _{i<j}G(z_i-z_j)-n\sum _{i=1}^{n}G(z_i).$$ A critical point $a=(a_1,\cdots ,a_n)$ of $G_n$ is called trivial if $\{a_1,\cdots ,a_n\}=\{-a_1,\cdots ,-a_n\}$. For such a point $a$, two geometric quantities $D(a)$ and $H(a)$ arising from bubbling analysis of mean field equations are introduced. $D(a)$ is a global quantity measuring asymptotic expansion and $H(a)$ is the Hessian of $G_n$ at $a$. By way of geometry of Lamé curves developed in our previous paper (Cambridge J. Math 3, 2015), we derive precise formulas to relate these two quantities.

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