On complete spacelike hypersurfaces with two distinct principal curvatures in a de Sitter space

Abstract

We investigate complete spacelike hypersurfaces in a de Sitter space with two distinct principal curvatures and constant m-th mean curvature. By using Otsukiʼs idea, we obtain some global classification results. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in de Sitter ( n + 1 ) -spaces S 1 n + 1 ( 1 ) ( n ⩾ 3 ) of constant m-th mean curvature H m ( | H m | ⩾ 1 ) with two distinct principal curvatures λ and μ satisfying inf ( λ − μ ) 2 > 0 are the hyperbolic cylinders. We also obtain some global rigidity results for hyperbolic cylinders and obtain some non-existence results.