Complete minimal hypersurfaces in ${\mathbb{H}}^5$ with constant scalar curvature and zero Gauss-Kronecker curvature

Abstract

We show that any complete minimal hypersurface in the five-dimensional hyperbolic space ${\mathbb{H}}^5$, endowed with constant scalar curvature and vanishing Gauss-Kronecker curvature, must be totally geodesic. Cheng-Peng [3] recently conjecture that any complete minimal hypersurface with constant scalar curvature in ${\mathbb{H}}^4$ is totally geodesic. Our result partially confirms this conjecture in five dimensional setting.

11 pages, no figure