Degenerate neckpinches in Ricci flow

TL;DR

This paper proves the existence of Ricci flow solutions that develop Type-II degenerate neckpinch singularities, matching previously derived asymptotic profiles, across various dimensions and singularity rates.

Contribution

It establishes the existence of Ricci flow solutions with specific degenerate neckpinch singularities for all dimensions and rates, confirming prior formal asymptotic predictions.

Findings

Existence of Ricci flow solutions with Type-II degenerate neckpinches.

Solutions develop singularities at rate (T-t)^{-2+2/k} for each integer k ≥ 3.

Results hold in all dimensions m ≥ 3.

Abstract

In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer $k\geq3$, there exist compact solutions in all dimensions $m\geq3$ that become singular at the rate (T-t)^{-2+2/k}$.