Stable CMC and index one minimal surfaces in conformally flat manifolds

TL;DR

This paper classifies stable constant mean curvature and index one minimal surfaces in conformally flat 3-manifolds with nonnegative Ricci curvature, showing they are topologically spheres or tori with specific geometric properties.

Contribution

It proves that such surfaces are topologically spheres or tori, and characterizes the torus case as embedded, minimal, and conformal to a flat square torus, extending understanding of minimal surfaces in these manifolds.

Findings

Stable CMC and index one minimal surfaces are topologically spheres or tori.

Tori are embedded, minimal, and conformal to flat square tori with zero Ricci curvature in the normal direction.

In conformally flat 3-spheres with positive Ricci curvature, isoperimetric domains are topological 3-balls.

Abstract

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $\Sigma$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $\Sigma$ is homeomorphic either to a sphere or to a torus. Moreover, in case $\Sigma$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit field normal to $\Sigma.$ The result is sharp, we can perturb the standard metric on the 3-sphere in its conformal class to obtain metrics of nonnegative Ricci curvature admitting minimal tori which are stable as CMC surfaces. As a consequence, in any 3-sphere of positive Ricci curvature which is conformally flat, the isoperimetric domains are topologically 3-balls. This proves a special case of a conjecture of A. Ros.