On the parabolicity of linear Weingarten spacelike submanifolds in the de sitter space

Abstract

In this paper, we deal with complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector in de Sitter space  S_p^n+p_. of index p. Imposing appropriate restrictions on the values of the mean cur-vature function H, we establish a parabolicity criterion related to a suitable Cheng-Yau's modied operator and we use it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be either  totally umbilical or isometric to certain hyperbolic cylinders of S_p^n+p_.