Centroaffine minimal hypersurfaces in \(\mathbb{R}^{n+1}\)
type: Item
instance of: scholarly article
title: Centroaffine minimal hypersurfaces in \(\mathbb{R}^{n+1}\)
zbMATH Open document ID: 0827.53010
DOI: 10.1007/BF01264101
published in: Geometriae Dedicata
publication date: 1994-10-18T00:00:00Z
review text: If a hypersurface \(M\) in \(\mathbb{R}^{n + 1}\) is transversal, i.e., if the position vector field \(x\) is nowhere tangent to \(M\), then one can consider \(x\) as affine normal, usually called the centroaffine normal. Then a symmetric bilinear 2-form \(g\) is defined on \(M\) by considering the second fundamental form of \(M\) with respect to the affine normal \(x\). If this form \(g\) is nondegenerate, then it is called the centroaffine metric and \(M\) is called a centroaffine hypersurface. Since it makes no sense to define the centroaffine mean curvature as the trace of the affine shape operator (which is always the identity, up to sign), one has to define another endomorphism as centroaffine shape operator, which is explained in the first part of this paper. In the second part, it is proved that this definition is good, in the sense that if one considers the area functional, determined by the centroaffine metric, then a hypersurface has critical area if and only if its centroaffine mean curvature vanishes. In the third part the second variation of the area is computed, and it is proved that a hyperbolic equiaffine hypersurface, considered as centroaffine minimal hypersurface, is stable and that an ellipsoid is unstable.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 663734
zbMATH Keywords: stability
zbMATH Keywords: centroaffine hypersurface
zbMATH Keywords: critical area
zbMATH Keywords: centroaffine mean curvature
zbMATH Keywords: centroaffine minimal hypersurface
author: Changping Wang
reviewed by: Franki Dillen
MaRDI profile type: Publication
cites work: Hypersurfaces with Maximal Affinely Invariant Area
cites work: Variational formulas for higher affine mean curvatures
cites work: The Affine Mean Curvature Vector for Surfaces in R4
cites work: Canonical centroaffine hypersurfaces in \({\mathbb{R}{}}^{n+1}\)
cites work: Q5288993
cites work: Affine variation formulas and affine minimal surfaces
cites work: Variation of curvature integrals
cites work: Centroaffine differential geometry: Submanifolds of codimension 2
cites work: Compact centroaffine spheres of codimension 2
full work available at URL: https://doi.org/10.1007/bf01264101
OpenAlex ID: W2039498654
description: scientific article; zbMATH DE number 663734