Complete surfaces of constant anisotropic mean curvature

This research has been financially supported by: Projects PID2020-118137GB-I00 and CEX2020- 001105-M, funded by MCIN/AEI /10.13039/501100011033; CARM, Programa Regional de Fomento de la Investigación, Fundación Séneca-Agencia de Ciencia y Tecnología Región de Murcia, reference 21937/PI/22; and Project 88881.133043/2016-01, funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES).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.MCIN/AEI /10.13039/501100011033 PID2020-118137GB-I00, CEX2020- 001105-MCARMFundación Séneca-Agencia de Ciencia y Tecnología Región de Murcia 21937/PI/22, 88881.133043/2016-01Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES