Hypersurfaces with constant mean curvature in hyperbolic space form

Abstract

In this article, we prove the following theorem: A complete hypersurface of the hyperbolic space form, which has constant mean curvature and non-negative Ricci curvature Q, has non-negative sectional curvature. Moreover, if it is compact, it is a geodesic distance sphere; if its soul is not reduced to a point, it is a geodesic hypercylinder; if its soul is reduced to a point p, its curvature satisfies ∥▽Q∥<∞, and the geodesic spheres centered at p are convex, then it is a horosphere.