Anisotropic umbilic points and Hopf's Theorem for surfaces with constant anisotropic mean curvature

TL;DR

This paper proves that topological spheres with constant anisotropic mean curvature are rescaled Wulff shapes, under conditions on the elliptic functional and Wulff shape.

Contribution

It establishes a Hopf-type theorem for anisotropic mean curvature surfaces, characterizing spheres as rescaled Wulff shapes.

Findings

Surfaces with constant anisotropic mean curvature and spherical topology are rescaled Wulff shapes.

The result applies to elliptic functionals with smooth, positively curved Wulff shapes.

Provides a geometric characterization analogous to classical results for isotropic mean curvature.

Abstract

We show that for elliptic parametric functionals whose Wulff shape is smooth and has strictly positive curvature, any surface with constant anisotropic mean curvature which is a topological sphere is a rescaling of the Wulff shape.