New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino and Henrique F. de Lima

Address:
C.P. Aquino, Departamento de Matemática, Universidade Federal do Piauí, 64049-550 Teresina, Piauí, Brazil
Corresponding author: H.F. de Lima, Departamento de Matemática, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil

E-mail:
cicero.aquino@ufpi.edu.br
henrique@dme.ufcg.edu.br

Abstract: In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb{H}^{n+1}$, that is, complete hypersurfaces of $\mathbb{H}^{n+1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R=aH+b$ for some $a$, $b\in \mathbb{R}$. In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb{H}^{n+1}$. Furthermore, a rigidity result concerning the compact case is also given.

AMSclassification: primary 53C42; secondary 53A10, 53B30, 53C50.

Keywords: hyperbolic space, linear Weingarten hypersurfaces, totally umbilical hypersurfaces, hyperbolic cylinders.

DOI: 10.5817/AM2015-4-201