New characterizations of linear Weingarten spacelike hypersurfaces in the de Sitter space

Luis J. Alías, Henrique F. de Lima and Fábio R. dos Santos

Vol. 292 (2018), No. 1, 1–19

DOI: 10.2140/pjm.2018.292.1

Abstract

We deal with complete linear Weingarten spacelike hypersurfaces immersed in the de Sitter space, that is, spacelike hypersurfaces of de Sitter space whose mean and scalar curvatures are linearly related. In this setting, we apply a suitable extension of the generalized maximum principle of Omori–Yau to show that either such a spacelike hypersurface must be totally umbilical or there holds a sharp estimate for the norm of its total umbilicity tensor, with equality characterizing hyperbolic cylinders of de Sitter space. We also study the parabolicity of these spacelike hypersurfaces with respect to a Cheng–Yau modified operator.

Keywords

de Sitter space, linear Weingarten hypersurfaces, spacelike hypersurfaces, totally umbilical hypersurfaces, hyperbolic cylinders, parabolicity

Mathematical Subject Classification 2010

Primary: 53C42

Secondary: 53A10, 53C20, 53C50

Milestones

Received: 6 December 2016

Accepted: 1 July 2017

Published: 22 September 2017