Complete surfaces of constant anisotropic mean curvature

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 $W$.

22 pages, 3 figures