Complete surfaces of constant anisotropic mean curvature

TL;DR

This paper classifies complete surfaces with constant anisotropic mean curvature in three-dimensional space, identifying conditions under which they are planes, cylinders, or homothetic to the Wulff shape, based on geometric and symmetry properties.

Contribution

It provides a comprehensive classification of complete CAMC surfaces under various geometric and symmetry assumptions, extending understanding of anisotropic surface geometry.

Findings

Planes and CAMC cylinders with Gauss map in a hemisphere are the only such surfaces.

Surfaces with non-zero CAMC and sign-consistent Gaussian curvature are either CAMC cylinders or Wulff shapes.

Surfaces with certain symmetry and topology are homothetic to the Wulff shape.

Abstract

We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only complete surfaces with CAMC whose Gauss map image is contained in a closed hemisphere of $\mathbb{S}^2$; (2) Any complete surface with non-zero CAMC and whose Gaussian curvature does not change sign is either a CAMC cylinder or the Wulff shape, up to a homothety of $\mathbb{R}^3$; and (3) if the Wulff shape $W$ of the anisotropic functional is invariant with respect to three linearly independent reflections in $\mathbb{R}^3$, then any properly embedded surface of non-zero CAMC, finite topology and at most one end is homothetic to $W$.