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

TL;DR

This paper extends Hopf-type theorems to surfaces in de Sitter-Schwarzschild and Reissner-Nordstrom manifolds, broadening classical results to include important solutions of Einstein's equations using advanced PDE techniques.

Contribution

It introduces new methods to prove Hopf-type theorems for surfaces in warped product manifolds related to general relativity, including de Sitter-Schwarzschild and Reissner-Nordstrom spaces.

Findings

Classifies constant mean curvature surfaces in these manifolds.

Extends classical Hopf theorems to new geometric contexts.

Uses PDE techniques generalizing holomorphy in complex analysis.

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. 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, using techniques partial differential equations in the complex plane which generalizes the notion of holomorphy, 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 the general relativity.