I will discuss a geometric invariant, that we call the width, of a manifold and first show how it can be realized as the sum of areas of minimal 2-spheres. 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 will estimate the rate of change of width under various geometric flows to prove sharp estimates for extinction times. This is joint work with Toby Colding.