The Morse property of limit functions appearing in mean field equations on surfaces with boundary

TL;DR

This paper investigates the Morse property of functions related to limit solutions of mean field equations on surfaces with boundary, showing that such functions can be made Morse through small conformal metric perturbations.

Contribution

It proves that for any Riemannian metric, a nearby conformal metric exists making the associated function Morse, and that this property is generic when all coefficients are positive.

Findings

Existence of conformal metrics making the function Morse

Openness and density of metrics with Morse functions when all coefficients are positive

Applicability to mean field equations on surfaces with boundary

Abstract

In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $\Sigma$ with boundary $\partial\Sigma$. Given a Riemannian metric $g$ on $\Sigma$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^m\sigma_i^2R^g(x_i)+\sum_{i,j=1\\i\ne j}^m\sigma_i\sigma_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $\sigma_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(\Sigma,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(\Sigma^m,\mathbb{R})$ is arbitrary. We prove that for any Riemannian metric $g$, there exists a metric $\widetilde g$ which is arbitrarily close to $g$ and in the conformal class of $g$ such that $f_{\widetilde g}$ is a Morse function. Furthermore we show that, if all $\sigma_i>0$, then the set of Riemannian metrics for which $f_g$ is a Morse function is open and dense in the set of all Riemannian metrics.