Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature

Abstract

We construct spherical space forms $\left(S^3∕\mathrm{\Gamma },g\right)$ with positive scalar curvature and containing no stable embedded minimal surfaces such that the following happens along the Ricci flow starting at $\left(S^3∕\mathrm{\Gamma },g\right)$: a stable embedded minimal $2$–sphere appears and a nontrivial singularity occurs. We also give in dimension $3$ a general construction of Type I neckpinching and clarify the relationship between stable spheres and nontrivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of singularities which can occur for metrics with these symmetries.