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- W4301097908 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." @default.
- W4301097908 created "2022-10-04" @default.
- W4301097908 creator A5005900826 @default.
- W4301097908 date "2013-06-19" @default.
- W4301097908 modified "2023-10-16" @default.
- W4301097908 title "Stable CMC and index one minimal surfaces in conformally flat manifolds" @default.
- W4301097908 doi "https://doi.org/10.48550/arxiv.1306.4458" @default.
- W4301097908 hasPublicationYear "2013" @default.
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