Geometry & Topology Volume 12, issue 5 (2008)

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

Primary: 53C42, 53C44

Secondary: 58E12, 58E20

References

Publication

Received: 30 June 2007
Accepted: 10 October 2008
Published: 6 November 2008
Proposed: Ben Chow
Seconded: Colin Rourke, Martin Bridson

Authors

Tobias H Colding
Department of Mathematics, MIT 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA and Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA

William P Minicozzi II
Department of Mathematics Johns Hopkins University 3400 N Charles St Baltimore, MD 21218 USA