{"title":"The Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary","authors":"Zhengni Hu, Thomas Bartsch","doi":"10.1007/s12220-024-01664-z","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface <span>\\(\\Sigma \\)</span> with boundary <span>\\(\\partial \\Sigma \\)</span>. Given a Riemannian metric <i>g</i> on <span>\\(\\Sigma \\)</span> we consider functions of the form\n</p><p>where <span>\\(\\sigma _i \\ne 0\\)</span> for <span>\\(i=1,\\ldots ,m\\)</span>, <span>\\(G^g\\)</span> is the Green function of the Laplace-Beltrami operator on <span>\\((\\Sigma ,g)\\)</span> with Neumann boundary conditions, <span>\\(R^g\\)</span> is the corresponding Robin function, and <span>\\(h \\in {{\\mathcal {C}}}^{2}(\\Sigma ^m,\\mathbb {R})\\)</span> is arbitrary. We prove that for any Riemannian metric <i>g</i>, there exists a metric <span>\\(\\widetilde{g}\\)</span> which is arbitrarily close to <i>g</i> and in the conformal class of <i>g</i> such that <span>\\(f_{\\widetilde{g}}\\)</span> is a Morse function. Furthermore we show that, if all <span>\\(\\sigma _i>0\\)</span>, then the set of Riemannian metrics for which <span>\\(f_g\\)</span> is a Morse function is open and dense in the set of all Riemannian metrics.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01664-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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
where \(\sigma _i \ne 0\) 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.