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

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.

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width, sweepout, min-max, Ricci flow, extinction, harmonic map, bubble convergence
Mathematical Subject Classification 2000
Primary: 53C44, 53C42
Secondary: 58E12, 58E20
Received: 30 June 2007
Accepted: 10 October 2008
Published: 6 November 2008
Proposed: Ben Chow
Seconded: Colin Rourke, Martin Bridson
Tobias H Colding
Department of Mathematics, MIT
77 Massachusetts Avenue
Cambridge, MA 02139-4307, USA
Courant Institute of Mathematical Sciences
251 Mercer Street
New York, NY 10012
William P Minicozzi II
Department of Mathematics
Johns Hopkins University
3400 N Charles St
Baltimore, MD 21218