A note on the stability of the Wulff shape

Abstract

We give a new proof of Palmer's result [6] that the Wulff shapes are the only closed, oriented, stable hypersurfaces with constant anisotropic mean curvature. Our approach is based on the construction of a suitable testfunction in the anisotropic index form, thus generalizing the original proof of Barbosa, do Carmo [1].

Anisotropic isoparametric hypersurfaces in Euclidean spaces

“…As a generalization of the usual area functional, extensive research has been devoted to study the isoperimetric problem and variational problem in anisotropic version which naturally introduce the notions of Wulff shape and the F-Weingarten (F-shape) operator as generalizations of unit sphere and the Weingarten (shape) operator of a hypersurface in a Euclidean space (cf. [1,[4][5][6]8,13,14,24,26], etc.). Due to the abundance of anisotropic surface energy functionals, the anisotropic property of Wulff shapes, and most troublesomely the asymmetry of the F-Weingarten operators, it turns out that generalizations of standard results to anisotropic version are not trivial and sometimes even impossible because of the great difference.…”

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

“…We consider the map (2) φ : S n → R n+1 , x → F (x)x + (grad S n F ) x , its image W F = φ(S n ) is a smooth, convex hypersurface in R n+1 called the Wulff shape of F (see [2], [3], [15], [10], [11], [12], [13], [17], [22], [23]). When F ≡ 1, the Wulff shape W F is just S n .…”

On surfaces of prescribed weighted mean curvature

“…As our problems do not necessarily arise as Euler equations of variational problems, we do not have the tool of second variation at hand. Instead, we have used only geometric identities to derive our equation for the normal, similarly to Winklmann in [16,Theorem 3.1], where the case of constant weighted mean curvature is studied.…”

“…We denote by N : Ω → S n its normal vector. In recent papers such as [4,14,15] or [16] critical points of the specific, anisotropic parametric functional…”