Width and finite extinction time of Ricci flow

Understand

This is an expository article with complete proofs intended for a general non-specialist 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'' $M$.