Birkhoff minimax principle for minimal surfaces with a free boundary
type: Item
instance of: scholarly article
title: Birkhoff minimax principle for minimal surfaces with a free boundary
zbMATH Open document ID: 0938.58015
author: Goufang Wang
published in: Mathematische Annalen
publication date: 1999-09-23T00:00:00Z
review text: The author proves a multiplicity result on the number of nonconstant minimal surfaces with a free boundary on a convex hypersurface \(S\) in \(\mathbb{R}^{n+1}\). He shows that if \(S\) satisfies the following pinching condition: there exists a diffeomorphism \(f:(S^n,g_0) \to(S,g)\), with \(S^n\) unit sphere in \(\mathbb{R}^{n+1}\) and \(g_0\) the standard metric on \(S^n\), such that \(g_0\leq f^*(g)\leq 2g_0\) \((f^*(g)\) is the pull-back metric on \(S^n)\), then there exist at least \(l(n)\) distinct nonconstant minimal surfaces with a free boundary on \(S\), where \(l(n)= n+2^k-1\), \(2^k\leq n<2^{k+1}\) is the cuplength of the Grassmannian \(G(n+1,2)\). The proof is based on a perturbation method used by several authors to study harmonic maps between Riemannian manifolds. The perturbation method is needed since the Palais-Smale condition for the variational problem associated is not satisfied. The author develops also a Birkhoff min-max theory for perturbed functionals.
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1337026
zbMATH Keywords: Birkhoff principle
zbMATH Keywords: free boundary problem
zbMATH Keywords: minimal surfaces
reviewed by: Antonio Masiello
MaRDI profile type: Publication
full work available at URL: s002080050287
OpenAlex ID: W2064155102
DOI: 10.1007/S002080050287
description: scientific article; zbMATH DE number 1337026