Precise asymptotics of the Ricci flow neckpinch

Abstract

The best known finite-time local Ricci flow singularity is the neckpinch, in which a proper subset of the manifold becomes geometrically close to a portion of a shrinking cylinder. In this paper, we prove precise asymptotics for rotationally-symmetric Ricci flow neckpinches. We then compare these rigorous results with formal matched asymptotics for fully general neckpinch singularities.

Neck Pinching Dynamics under Mean Curvature Flow

“…The previous result closest to our result is that by Angenent and Knopf [4,3] on the neckpinching for the Ricci flow of SO(n + 1)− invariant metrics on S n+1 .…”

Formal matched asymptotics for degenerate Ricci flow neckpinches

“…Use of τ is not necessary but is convenient for the calculations that follow. (Compare [3].) The evolution equation satisfied by ψ, equation (2.6b), implies that at the local maximum ("bump")ŝ(t) closest to the right pole, one has ψ t (ŝ, t) ≤ (n − 1)/ψ.…”

“…(4) The metric has at least one neck and is "sufficiently pinched" in the sense that the value of ψ at the smallest neck is sufficiently small relative to its value at either adjacent bump. In [3], precise asymptotics are derived under the additional assumption:…”

“…In particular, we drop the condition on the "tightness" of the initial neck pinching, and we also do not require reflection symmetry. 3 We now state our underlying assumptions on the initial geometry:…”

Type II Extinction Profile of Maximal Solutions to the Ricci Flow in ℝ2

“…The precise asymptotics of the Ricci flow neckpinch in the compact case, on S n , has been established by Knopf and Angenent in [2].…”