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

Song, Antoine

doi:10.2140/gt.2019.23.3501

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

Antoine Song

Geometry & Topology 23 (2019) 3501–3535

DOI: 10.2140/gt.2019.23.3501

Abstract

We construct spherical space forms $(S^{3} ∕ Γ, g)$ with positive scalar curvature and containing no stable embedded minimal surfaces such that the following happens along the Ricci flow starting at $(S^{3} ∕ Γ, g)$ : 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.

Keywords

Ricci flow, minimal spheres, scalar curvature

Mathematical Subject Classification 2010

Primary: 53A10, 53C44

References

Publication

Received: 3 May 2018

Revised: 8 January 2019

Accepted: 9 January 2019

Published: 30 December 2019

Proposed: John Lott

Seconded: Ian Agol, Bruce Kleiner

Authors

Antoine Song
Department of Mathematics Princeton University Princeton, NJ United States

Volume 23, issue 7 (2019)