Sphere-Foliated Minimal and Constant Mean Curvature Hypersurfaces in Space Forms and Lorentz-Minkowski Space

Abstract

We prove that a sphere-foliated minimal or constant mean curvature hypersurface in hyperbolic space of dimension > 5 is one of the following: hypersurface of rotation around a geodesic, geodesic hyperplane, horosphere, equidistant hypersurface, or a geodesic sphere in the upper half-space model. And we show that a sphere-foliated minimal or constant mean curvature hypersurface in sphere of dimension > 5 is either a hypersurface of rotation or a hypersphere. We also show that a hypersurface of nonzero constant mean curvature in Lorentz-Minkowski space foliated by spheres in space-like hyperplanes is either a hypersurface of rotation or a pseudo-hyperbolic space and that maximal space-like hypersurfaces foliated by spheres in hyperplanes are rotational if the ambient space has dimension > 4.