The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres

TL;DR

This paper investigates the behavior of Ricci flow on noncompact 3-manifolds with essential minimal spheres, revealing conditions under which singularities form and providing numerical insights into scalar curvature effects.

Contribution

It extends understanding of Ricci flow dynamics on noncompact manifolds with minimal spheres, especially in intermediate cases with scalar curvature constraints.

Findings

Finite-time singularity formation under certain scalar curvature conditions

Numerical exploration of scalar curvature effects on Ricci flow

Application to asymptotically flat manifolds and boundary cases

Abstract

It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.