The Morse property of limit functions appearing in mean field equations on surfaces with boundary
type: Item
instance of: scholarly article
title: The Morse property of limit functions appearing in mean field equations on surfaces with boundary
zbMATH Open document ID: 1540.3513
author: Zhengni Hu
author: Thomas J. Bartsch
published in: The Journal of Geometric Analysis
publication date: 2024-05-13T00:00:00Z
review text: Let \(\Sigma\) denote a smooth and compact surface with boundary \(\partial \Sigma\) of a Riemannian manifold with metric \(g\). Denote also by \(G^g\) the Green function of the Laplace-Beltrami operator with Neuman boundary condition.\N\NThis article discusses Morse properties of functions of the form \N\[\Nf_g(x)=\sum_{1\leq i\leq m} \sigma_i^2 R^g(x_i)+\sum_{1\leq i, j\leq m, i\neq j}\sigma_i \sigma_j G^g(x_i, x_j)+h(x_1, x_2, \dots, x_m), \N\]\Nwhere \(R^g\) is the corresponding Robin function of \(G^g\) and \(h\) is any \(C^2\) arbitrary function. The main result of the article establishes that for any Riemannian metric \(g\), there exists a metric \(\tilde g\) arbitrarily close to \(g\) and in the conformal class of with the property that \(f_{\tilde g}\) is a is a Morse function. As a byproduct, it is obtained that if all \(\sigma_i>0\), then the set of all Riemannian metrics \(g\) for which \(f_g\) is a Morse function becomes an open and dense set in the set of all Riemannian metrics.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 35J08
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 7846478
zbMATH Keywords: Green function
zbMATH Keywords: Robin function
zbMATH Keywords: Morse property
zbMATH Keywords: transversality theorem
MaRDI profile type: Publication
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DOI: 10.1007/S12220-024-01664-Z
description: scientific article; zbMATH DE number 7846478