Hopf type theorems in Riemannian manifolds

Abstract

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize the result of Hopf as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

26 pages. Survey article about the contributions of Renato Tribuzy and the authors to the study of the Hopf's differential of surfaces immersed in Riemannian manifolds. Accepted to be published in a special volume of the Matemática Contemporânea, a journal of the Brazilian Mathematical Society, in honor of Renato de Azevedo Tribuzy on the occasion of his 75th birthday