Hopf type theorems for surfaces in the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds

Abstract

In 1951, Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in Euclidean space are the round (geometrical) spheres. These results were generalized by S. S. Chern and then by Eschenburg and Tribuzy for surfaces homeomorphic to the sphere in Riemannian manifolds with constant sectional curvature whose mean curvature function satisfies some bound on its differential. In this paper, we extend these results for surfaces in a wide class of warped product manifolds, which includes, besides the classical space forms of constant sectional curvature, the de Sitter–Schwarzschild manifolds and the Reissner–Nordstrom manifolds, which are time slices of solutions of the Einstein field equations of general relativity.