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

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.

v2: title changed, first part will appear elsewhere, second part about appearance of stable spheres is extended substantially

References in corpus (10)

• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Minimal hypersurfaces with bounded index
• Compactness of minimal hypersurfaces with bounded index
• Long-time behavior of 3 dimensional Ricci flow -- Introduction
• Existence of minimal surfaces of arbitrary large Morse index
• Metrics of positive scalar curvature and unbounded widths
• Long-time behavior of 3 dimensional Ricci flow -- A: Generalizations of Perelman's long-time estimates
• Long-time behavior of 3 dimensional Ricci flow -- D: Proof of the main results
• Long-time behavior of 3 dimensional Ricci flow -- B: Evolution of the minimal area of simplicial complexes under Ricci flow
• Long-time behavior of 3 dimensional Ricci flow -- C: 3-manifold topology and combinatorics of simplicial complexes in 3-manifolds

Cited by in corpus (1)

• Local min-max surfaces and strongly irreducible minimal Heegaard splittings