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

TL;DR

This paper constructs examples of 3D manifolds with positive scalar curvature where stable minimal spheres appear during Ricci flow, linking their emergence to specific singularity types and symmetry conditions.

Contribution

It provides explicit examples of spherical space forms with no initial stable minimal surfaces that develop stable spheres and singularities under Ricci flow, and clarifies the role of symmetry.

Findings

Stable minimal spheres can appear during Ricci flow in positive scalar curvature manifolds.

Symmetry assumptions can prevent the formation of stable spheres.

The work links stable spheres to Type I singularities in Ricci flow.

Abstract

We construct examples of spherical space forms $(S^3/\Gamma,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/\Gamma,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension $3$ a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial 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.