Pseudo-parallel immersions in SnxR and HnxR, and constant anisotropic mean curvature surfaces in R3 - Tassi, Marcos Paulo (2019)
In this Ph.D. thesis, we investigate two topics in Differential Geometry. The first topic refers to the study of pseudo-parallel submanifolds in the ambient spaces SnxR and HnxR. We complete the partial classification given by F. Lin and B. Yang. As a consequence, we classify minimal and constant mean curvature pseudo-parallel hypersurfaces. We also prove a characterization of pseudo-parallel surfaces in SnxR and HnxR, for n ≥ 4, and the non-existence of pseudo-parallel surfaces with non-vanishing normal curvature, when n=3. The second part of the thesis is devoted to the study of constant anisotropic mean curvature surfaces in R3. We obtain a Bernstein-type Theorem for multigraphs with constant anisotropic mean curvature, an anisotropic version of a theorem proved by D. Hoffman, R. Osserman and R. Schoen, in 1982. As a consequence, we prove that complete surfaces with non-zero constant anisotropic mean curvature and whose Gaussian curvature does not change sign are either the Wulff shape or cylinders. We prove uniform height estimates for vertical graphs with non-zero constant anisotropic mean curvature and planar boundary, a generalization of the theorem proved by W. Meeks, in 1988, and we obtain uniform height estimates for compact embedded surfaces with non-zero constant anisotropic mean curvature and planar boundary, as a corollary. We also prove, under certain symmetry hypothesis on the anisotropy function, the non-existence of properly embedded surfaces in R3 with non-zero constant anisotropic mean curvature and with just one end.