The Modelling of Degenerate Neck Pinch Singularities in Ricci Flow by Bryant Solitons

TL;DR

This paper investigates the behavior of Ricci flow near degenerate neck pinch singularities on S^3, providing numerical evidence that Bryant steady solitons model the flow at the poles.

Contribution

It demonstrates that Bryant steady solitons accurately model the polar flow in Ricci flow degenerating neck pinch singularities on S^3.

Findings

Numerical simulations support the model of degenerate neck pinch formation.

Bryant steady solitons effectively describe the flow at the poles during singularity formation.

The study confirms the conjecture about Bryant solitons modeling polar behavior in Ricci flow.

Abstract

In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point between initial geometries (on $S^3$) whose volume-normalized Ricci flows develop a singular neck pinch, and other initial geometries whose volume normalized Ricci flows converge to a round sphere - evolves into a ``degenerate neck pinch.'' That is, we have seen in this earlier work that the Ricci flows for the critical geometries become locally cylindrical in a neighborhood of the initial pinching, and have the maximum amount of curvature at one or both of the poles. Here, we explore the behavior of these flows at the poles, and find strong support for the conjecture that the Bryant steady solitons accurately model this polar flow.