Hopf type theorems in Riemannian manifolds

TL;DR

This paper surveys generalizations of Hopf's theorem on constant mean curvature surfaces, including recent advances in warped product manifolds and applications to curvature flows.

Contribution

It compiles and discusses extensions of Hopf's classical result to new geometric contexts and recent results involving shrinking solitons and specific warped product manifolds.

Findings

Generalizations of Hopf's theorem to warped product manifolds

Recent results on shrinking solitons of curvature flows

Applications to surfaces in de Sitter-Schwarzschild and Reissner-Nordstrom 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.