Complete hypersurfaces in a 4-dimensional hyperbolic space

Abstract

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H 4 (−1), whose scalar curvature is bounded from below. §1 Introduction Let M 3 be a 3-dimensional hypersurface in a simply connected space form F 4 (c) of constant sectional curvature c. Peng and Terng [1] and Cheng [2] proved that Cartan's isoparametric minimal hypersurfaces with zero Gauss-Kronecker curvature are the only complete minimal hypersurfaces in unit sphere S 4 (1) with constant scalar curvature and three distinct principal curvatures at each point. On the other hand, Almeida and Brito [3] considered the closed minimal hypersurfaces in S 4 (1) with constant Gauss-Kronecker curvature. They proved that the Clifford torus S 2 × S 1 and the boundary of the tube of a minimal 2-dimensional surface in S 4 (1) with geodesic radius π 2 are the only closed minimal hypersurfaces in S 4 (1) with constant GaussKronecker curvature and non-vanishing second fundamental form. Ramanathan [4] extended this result and allowed points where the second fundamental form is zero. Recently, Almeida, Brito and Sousa [5] completely classified the closed hypersurfaces of S 4 (1) where any two of the mean curvature, the scalar curvature and the Gauss-Kronecker curvature are constant. In [6], Hasanis, Savas-Halilaj and Vlachos extended the above results to complete minimal hypersurfaces in Euclidean space. It is natural to investigate complete hypersurfaces in hyperbolic space H n+1 (−1) of constant curvature −1 under some curvature conditions. Cheng [7] showed that the Gauss-Kronecker curvature of complete minimal hypersurfaces in H 4 (−1) with constant Gauss-Kronecker curvature must be zero if its scalar curvature is bounded from below. We know that the quasi-Gauss-Kronecker curvature is exactly the Gauss-Kronecker curvature Received: 2008-12-02 MR Subject Classification: 53C24, 53C40