Deformation and stability of surfaces with constant mean curvature
type: Item
instance of: scholarly article
title: Deformation and stability of surfaces with constant mean curvature
zbMATH Open document ID: 1010.58007
author: Miyuki Koiso
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2002-10-08T00:00:00Z
review text: From the introduction and abstract: An immersion \({\mathcal X}:M\to \mathbb{R}^3\) of a two-dimensional orientable compact connected \(C^\infty\) manifold \(M\) with boundary \(\partial M\) into the Euclidean three-space \(\mathbb{R}^3\) has constant mean curvature if and only if \({\mathcal X}\) is a critical point of the area functional for all volume-preserving variations of \({\mathcal X}\) that fix the boundary. When the mean curvature of \({\mathcal X}\) is constant (we will say that \({\mathcal X}\) is a CMc immersion), \({\mathcal X}\) is said to be stable if the second variation of the area functional is nonnegative for all such variations of \({\mathcal X}\) is above. For a CMC immersion from a two-dimensional compact smooth manifold with boundary into the Euclidean three-space, we give sufficient conditions under which it has a CMC deformation fixing the boundary. Moreover, we give a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of eigenvalues and eigenfunctions of an eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtain a `visible' way of judging the stability.
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1775139
zbMATH Keywords: CMC immersion
zbMATH Keywords: CMC deformation
zbMATH Keywords: stability
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: second variation
zbMATH Keywords: area functional
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113247184
OpenAlex ID: W2047308799
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DOI: 10.2748/TMJ/1113247184
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description: scientific article; zbMATH DE number 1775139