Width and finite extinction time of Ricci flow

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.