Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

TL;DR

This paper proves the existence of constant mean curvature surfaces in asymptotically flat and hyperbolic 3-manifolds using min-max theory and inverse mean curvature flow properties.

Contribution

It establishes the existence of such surfaces with prescribed mean curvature in these manifolds, combining geometric analysis techniques.

Findings

Existence of constant mean curvature surfaces in asymptotically flat manifolds for any positive mean curvature.

Existence of constant mean curvature surfaces in asymptotically hyperbolic manifolds for mean curvature greater than 2.

Inverse mean curvature flow remains smooth for all times when starting from a point far out in the manifold.

Abstract

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge 0$. Then, for each constant $c>0$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $\Sigma \subset M$ with mean curvature $c$. Likewise, let $(M^3,g)$ be an asymptotically hyperbolic manifold with scalar curvature $R\ge -6$. Then, for each constant $c>2$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $\Sigma \subset M$ with mean curvature $c$. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any $T$ the inverse mean curvature flow emerging out of a point $p$ far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times $t\in (-\infty,T]$.