Applications of Hamilton’s Compactness Theorem for Ricci flow

1 Contents Applications of Hamilton's Compactness Theorem for Ricci flow Peter Topping 1 Applications of Hamilton's Compactness Theorem for Ricci flow 3 Overview 3 Background reading 4 Lecture 1. Ricci flow basics – existence and singularities 5 1.1. Initial PDE remarks 5 1.2. Basic Ricci flow theory 6 Lecture 2. Cheeger-Gromov convergence and Hamilton's compactness theorem 9 2.1. Convergence and compactness of manifolds 9 2.2. Convergence and compactness of flows 11 Lecture 3. Applications to Singularity Analysis 13 3.1. The rescaled flows 13 3.2. Perelman's no local collapsing theorem 14 Lecture 4. The case of compact surfaces – an alternative approach to the results of Hamilton and Chow 17 Lecture 5. The 2D case in general – Instantaneously Complete Ricci flows 21 5.1. How to pose the Ricci flow in general 21 5.2. The existence and uniqueness theory 22 5.3. Asymptotics 24 5.4. Singularities not modelled on shrinking spheres 25 Lecture 6. Contracting Cusp Ricci flows 27 Lecture 7. Subtleties of Hamilton's compactness theorem 31 7.1. Intuition behind the construction 32 7.2. Fixing proofs requiring completeness in the extended form of Hamilton's compactness theorem 33 Bibliography 35 i Overview In these lectures, I will try to give an introduction to two separate aspects of Ricci flow, namely Hamilton's compactness theorem and the very neat theory of Ricci flow in 2D. The target audience consists of graduate students with some background in differential geometry and PDE theory. Hamilton's compactness theorem is an absolutely fundamental tool in the modern theory of Ricci flow. I will spend the early part of the course explaining what this result says – roughly that given an appropriate sequence of Ricci flows, one can pass to a subsequence and get smooth convergence to a limit Ricci flow. In order to make sense of that, we will first look at the details of what it means for a sequence of Ricci flows, or simply of Riemannian manifolds, to converge in the Cheeger-Gromov sense. We will not assume any prior knowledge of this notion, although it will be almost essential to have some basic prior knowledge of Riemannian geometry, including the basic idea of the Riemannian curvature tensor. I will then go on to illustrate the most basic application of the compactness theorem, as envisaged by Hamilton and realised fully by Perelman, by blowing up a singularity to obtain a limit ancient Ricci flow modelling the singularity. To do …