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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.06530 |
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Table of Contents:
- In this paper we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface $Σ$ with boundary $\partialΣ$. Given a Riemannian metric $g$ on $Σ$ we consider functions of the form \[ f_g(x) := \sum_{i=1}^mσ_i^2R^g(x_i)+\sum_{i,j=1\ı\ne j}^mσ_iσ_jG^g(x_i,x_j)+h(x_1,\ldots,x_m), \] where $σ_i \neq 0$ for $i=1,\ldots,m$, $G^g$ is the Green function of the Laplace-Beltrami operator on $(Σ,g)$ with Neumann boundary conditions, $R^g$ is the corresponding Robin function, and $h \in \mathcal{C}^{2}(Σ^m,\mathbb{R})$ 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 $σ_i>0$, 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.