Complete surfaces of constant anisotropic mean curvature

Abstract

We study the geometry of complete immersed surfaces in R3 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 S2; (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 R3; and (3) if the Wulff shape W of the anisotropic functional is invariant with respect to three linearly independent reflections in R3, then any properly embedded surface of non-zero CAMC, finite topology and at most one end is homothetic to W.