On the Morse index of minimal surfaces in \({\mathbb{R}}^ p\) with polygonal boundaries
type: Item
instance of: scholarly article
title: On the Morse index of minimal surfaces in \({\mathbb{R}}^ p\) with polygonal boundaries
zbMATH Open document ID: 0607.53004
DOI: 10.1007/BF01626395
author: Friedrich Sauvigny
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155094
review text: Let \(\Gamma \subset {\mathbb{R}}^ p\) (p\(\geq 3)\) denote a Jordan polygon with \(N+3\) (N\(\geq 1)\) vertices. Then the minimal surfaces x spanning \(\Gamma\) correspond to the critical points \(\overset\circ\tau\) of an analytic function \(\theta\) : \(T\to {\mathbb{R}}\) in N variables. This function was introduced by M. Shiffman and the regularity of \(\theta\) has been investigated by E. Heinz. Now we are interested in the correspondence of the second order: The second derivative of \(\theta\) is given by its Hessian M(\(\overset\circ\tau\)), the second variation of the minimal surface x is described by the Schwarz operator \(S_ x\). We first study the eigenvalue problem of this singular differential operator by variational methods. Counting the negative eigenvalues of \(S_ x\) we obtain the Morse index \(m_ x\) of x. As central result we prove: The Morse index \(m_ x\) of the minimal surface x coincides with the Morse index m(\(\overset\circ\tau\)) of \(\theta\) (the number of negative eigenvalues of M(\(\overset\circ \tau\))) at the corresponding critical point \(\overset\circ\tau\in T\). This is achieved by comparison of the finite and infinite dimensional eigenvalue problems of M(\(\overset\circ\tau\)) and \(S_ x\), using certain simultaneous variations of the minimal surface.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 3981977
zbMATH Keywords: Jordan polygon
zbMATH Keywords: second variation
zbMATH Keywords: minimal surface
zbMATH Keywords: Schwarz operator
zbMATH Keywords: variational methods
zbMATH Keywords: Morse index
MaRDI profile type: Publication
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OpenAlex ID: W2319643056
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description: scientific article; zbMATH DE number 3981977