Preface Trinity: It's the question that drives us, Neo. It's the question that brought you here. You know the question just as I did. Neo: What is the Matrix? ... Morpheus: Do you want to know what it is? The Matrix is everywhere.... Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. - From the movie "The Matrix". What this book is about This is the sequel to the book "The Ricci Flow: An Introduction" by two of the authors [108]. In the previous volume (henceforth referred to as Volume One) we laid some of the foundations for the study of Richard Hamilton's Ricci flow. The Ricci flow is an evolution equation which deforms Riemannian metrics by evolving them in the direction of minus the Ricci tensor. It is like a heat equation and tries to smooth out initial metrics. In some cases one can exhibit global existence and convergence of the Ricci flow. A striking example of this is the main result presented in Chapter 6 of Volume One: Hamilton's topological classification of closed 3-manifolds with positive Ricci curvature as spherical space forms. The idea of the proof is to show, for any initial metric with positive Ricci curvature, the normalized Ricci flow exists for all time and converges to a constant curvature metric as time approaches infinity. Note that on any closed 2-dimensional manifold, the normalized Ricci flow exists for all time and converges to a constant curvature metric. Many of the techniques used in Hamilton's original work in dimension 2 have influenced the study of the Ricci flow in higher dimensions. In this respect, of special note is Hamilton's 'meta-principle' of considering geometric quantities which either vanish or are constant on gradient Ricci solitons. It is perhaps generally believed that the Ricci flow tries to make met- rics more homogeneous and isotropic. However, for general initial metrics on closed manifolds, singularities may develop under the Ricci flow in di- mensions as low as 3.1 In Volume One we began to set up the study of singularities by discussing curvature and derivative of curvature estimates, looking at how generally dilations are done in all dimensions, and studying For noncompact manifolds, finite time singularities may even occur in dimension 2. ix
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