Stable CMC and index one minimal surfaces in conformally flat manifolds

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.