The modeling of degenerate neck pinch singularities in Ricci flow by Bryant solitons

Abstract

In earlier work, carrying out numerical simulations of the Ricci flows 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 S3) 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.