Complete surfaces of constant anisotropic mean curvature

2019/12/04 by José A. Gálvez, Pablo Mira, Galvez, Jose A. +3
Mathematics · #53A10 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1912.01941

openalex publication_date 2019/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbS²; (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.

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