{"title":"有边界曲面上均值场方程中出现的极限函数的莫尔斯特性","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":"{\"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}","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
摘要
在本文中,我们研究了与边界为(部分)的光滑紧凑曲面()上均值场方程的极限函数相关的函数的莫尔斯性质。给定一个关于\(\西格玛\)的黎曼度量g,我们考虑的函数形式为:\(\西格玛_i \ne 0\) for \(i=1,\ldots ,m\),\(G^g\)是拉普拉斯-贝尔特拉米算子在\((\西格玛...g)\)上的格林函数、g))上具有诺伊曼边界条件的格林函数,(R^g)是相应的罗宾函数,(h in {{\mathcal {C}}}^{2}(\Sigma ^m,\mathbb {R})\) 是任意的。我们证明,对于任何黎曼度量 g,都存在一个度量 \(\widetilde{g}\),它任意地接近于 g 并且在 g 的共形类中,这样 \(f_{\widetilde{g}}\)就是一个莫尔斯函数。此外,我们还证明了,如果所有的 \(sigma_i>0\)都是莫尔斯函数,那么 \(f_g\)是莫尔斯函数的黎曼度量集合是开放的,并且在所有黎曼度量集合中是密集的。
The Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary
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.