Width and finite extinction time of Ricci flow

Colding, Tobias H¹ ; Minicozzi II, William P²

¹ 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
² Department of Mathematics, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA

Geometry & topology, Tome 12 (2008) no. 5, pp. 2537-2586.

Résumé

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.

Détail

DOI : 10.2140/gt.2008.12.2537

Keywords: width, sweepout, min-max, Ricci flow, extinction, harmonic map, bubble convergence

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