The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds

TL;DR

This paper extends compactness results for minimal and constant mean curvature surfaces in Riemannian manifolds, showing under certain curvature conditions that families of such surfaces are compact.

Contribution

It generalizes Choi-Schoen's Compactness Theorem to H-surfaces with small mean curvature in manifolds with positive Ricci curvature and proves compactness of convex CMC hypersurfaces in 1/4-pinched manifolds.

Findings

Extension of compactness theorem to H-surfaces with small mean curvature

Compactness of convex CMC hypersurfaces in 1/4-pinched manifolds

Conditions on Ricci and mean curvature for compactness

Abstract

The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed compact H-surfaces in a Riemannian manifold with positive Ricci curvature (the mean curvature small depending on the Ricci curvature). Also, we prove that the space of convex embedded (fixed) constant mean curvature hypersurfaces in a simply connected 1/4-pinched manifold is compact.