Complete surfaces of constant anisotropic mean curvature

• Main Authors: Galvez, Jose A, Mira, Pablo, Tassi, Marcos P
• Format: Journal Article
• Language: English
• Published: 04-12-2019
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Differential Geometry
• DOI: 10.48550/arxiv.1912.01941

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 $W$.