Surfaces with maximal constant mean curvature

TL;DR

This paper establishes an upper bound on the mean curvature of certain surfaces in asymptotically flat manifolds with non-negative scalar curvature, enabling the construction of a weak CMC foliation.

Contribution

It introduces a method to bound and maximize constant mean curvature surfaces in specific geometric settings, extending the understanding of CMC foliations.

Findings

Existence of an upper bound on mean curvature for surfaces homologous to boundary components.

Construction of a maximizer for the constant mean curvature.

Development of a weak CMC foliation concept.

Abstract

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This bound allows us to find a maximizer for the constant mean curvature of a surface homologous to the inner boundary. With this maximizer at hand, we can construct an increasing family of sets with boundaries of increasing constant mean curvature. We interpret this familiy as a weak version of a CMC foliation.