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

TL;DR

The paper proves finite-time extinction for the Ricci flow with surgery on closed oriented 3-manifolds whose prime decomposition has no aspherical factors. It achieves this by a differential inequality for least-area disks spanning contractible loops, combined with a regularized curve shortening flow (lifted to M×S^1) to control their area. The argument extends from irreducible to general M via the surgery framework and topological finiteness results, giving a route toward the elliptization conjecture. The approach builds on Hamilton's minimal-disk ideas and the Altschuler–Grayson regularization.

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.