#### Volume 12, issue 5 (2008)

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Width and finite extinction time of Ricci flow

### Tobias H Colding and William P Minicozzi II

Geometry & Topology 12 (2008) 2537–2586
##### Abstract

This is an expository article with complete proofs intended for a general nonspecialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal $2$–spheres. For instance, when $M$ is a homotopy $3$–sphere, the width is loosely speaking the area of the smallest $2$–sphere needed to ‘pull over’ $M$. Second, we use this to conclude that Hamilton’s Ricci flow becomes extinct in finite time on any homotopy $3$–sphere.

##### Keywords
width, sweepout, min-max, Ricci flow, extinction, harmonic map, bubble convergence
##### Mathematical Subject Classification 2000
Primary: 53C44, 53C42
Secondary: 58E12, 58E20