Almost-rigidity and the extinction time of positively curved Ricci flows

TL;DR

This paper proves that Ricci flows with positive isotropic curvature that nearly reach their maximal extinction time are close to round, with applications to positively curved metrics on 3-spheres and projective 3-spaces.

Contribution

It establishes a near-rigidity result for Ricci flows with positive isotropic curvature approaching maximal extinction time, linking curvature conditions to geometric roundness.

Findings

Ricci flows with almost maximal extinction time are nearly round.

Positively curved metrics on S^3 and RP^3 with almost maximal width are nearly round.

Provides a quantitative link between curvature conditions and geometric shape.

Abstract

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.