CHAPTER 3 The Compactness Theorem for Ricci Flow

The compactness of solutions to geometric and analytic equations, when it is true, is fundamental in the study of geometric analysis. In this chapter we state and prove Hamilton’s compactness theorem for solutions of the Ricci flow assuming Cheeger and Gromov’s compactness theorem for Riemannian manifolds with bounded geometry (proved in Chapter 4). In Section 3 of this chapter we also give various versions of the compactness theorem for solutions of the Ricci flow. Throughout this chapter, quantities depending on the metric gk (or gk (t)) will have a subscript k; for instance, ∇k and Rm k denote the Riemannian connection and Riemannian curvature tensor of gk. Quantities without a subscript depend on the background metric g. Often we suppress the t dependence in our notation where it is understood that the metrics depend on time while being defined on a space-time set. Given a sequence of quantities indexed by {k} , when we talk about a subsequence, most of the time we shall still use the indices {k} although we should use the indices {jk} .