On the Size of a Ricci Flow Neckpinch via Optimal Transport

TL;DR

This paper uses optimal transport techniques to analyze Ricci flow neckpinch singularities on spheres, removing symmetry assumptions and confirming single-point pinching behavior.

Contribution

It introduces a novel application of optimal transport to Ricci flow, removing symmetry constraints and providing a new proof of neckpinch singularity formation.

Findings

Confirmed single-point pinching without symmetry assumptions

Applied optimal transport to Ricci flow analysis

Extended understanding of Ricci flow singularities

Abstract

In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $\mathbb{S}^{n+1}$. In particular, we recover their proof of 'single-point pinching' along the flow. Using the methods of optimal transportation, we are able to remove the assumption of reflection symmetry for the metric. Our argument relies on the heuristic for weak Ricci flow proposed by McCann-Topping which characterizes super solutions of the Ricci flow by the contractivity of diffusions.