Centroaffine minimal hypersurfaces in \(\mathbb{R}^{n+1}\)
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.
scientific article; zbMATH DE number 663734