Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

TL;DR

This paper proves that solutions to the Ricci flow with surgery on certain closed three-manifolds become extinct in finite time, extending understanding of Ricci flow behavior on these manifolds.

Contribution

It establishes finite extinction time for Ricci flow with surgery on three-manifolds without aspherical factors, using minimal disk and curve shortening flow techniques.

Findings

Ricci flow solutions become extinct in finite time on specified manifolds

Utilizes minimal disk argument and curve shortening flow regularization

Extends previous results on Ricci flow behavior in three dimensions

Abstract

Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.