Uniqueness of stable closed non-smooth hypersurfaces with constant anisotropic mean curvature

TL;DR

This paper proves that under certain smoothness and convexity conditions, the only stable closed hypersurfaces with constant anisotropic mean curvature are rescaled Wulff shapes, extending classical geometric concepts to anisotropic energies.

Contribution

It establishes the uniqueness of stable equilibrium hypersurfaces with constant anisotropic mean curvature as rescaled Wulff shapes under smoothness and convexity assumptions.

Findings

Stable equilibrium hypersurfaces are rescaled Wulff shapes.

Provides generalizations of Steiner's and Minkowski's formulas to anisotropic energies.

Shows uniqueness of minimizers for anisotropic energy among closed hypersurfaces.

Abstract

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hypersurfaces enclosing the same (n+1)-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. We prove that, if the anisotropic energy density function is twice continuously differentiable and convex, then any closed stable equilibrium hypersurface is (up to rescaling) the Wulff shape. We also give fundamental definitions, many examples, and generalizations of well-known concepts and formulas like Steiner's formula and Minkowski's formula to the anisotropic case.