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# Communications in Analysis and Geometry

## Volume 28 (2020)

### Number 6

### Geometric quantities arising from bubbling analysis of mean field equations

Pages: 1289 – 1313

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n6.a2

#### Authors

#### 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 $E^n$:\[G_n (z_1, \dotsc , z_n) := \sum_{i \lt j} G (z_i - z_j) - n \sum^n_{i=1} G (z_i) \: \textrm{.}\]A critical point $a = (a_1, \dotsc , a_n)$ of $G_n$ is called trivial if $\lbrace a_1, \dotsc , a_n \rbrace = \lbrace -a_1, \dotsc , -a_n \rbrace$. 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 [3], we derive precise formulas to relate these two quantities.

Received 14 September 2016

Accepted 12 March 2018

Published 2 December 2020