On the stability of the Wulff shape

TL;DR

This paper investigates the stability of the Wulff shape as the unique stable solution to a variational problem involving higher order anisotropic curvatures, under certain convexity conditions.

Contribution

It proves that, up to translations and scalings, the Wulff shape is the only stable closed hypersurface for a class of anisotropic curvature functionals.

Findings

Wulff shape is the unique stable critical point.

Stability holds up to translations and homotheties.

The result applies under specific convexity assumptions on the anisotropic function.

Abstract

Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for which these hypersurfaces are critical points and we prove that, up to translations and homotheties, the Wulff shape is the only stable closed hypersurface of the Euclidean space for this problem.