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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2405.06530 |
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| _version_ | 1866909197580369920 |
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| author | Hu, Zhengni Bartsch, Thomas |
| author_facet | Hu, Zhengni Bartsch, Thomas |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Morse property of limit functions appearing in mean field equations on surfaces with boundary Hu, Zhengni Bartsch, Thomas Differential Geometry Analysis of PDEs 53C21 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. |
| title | The Morse property of limit functions appearing in mean field equations on surfaces with boundary |
| topic | Differential Geometry Analysis of PDEs 53C21 |
| url | https://arxiv.org/abs/2405.06530 |