# Manual The Ricci flow: an introduction

Perelman, Grisha November 11, The entropy formula for the Ricci flow and its geometric applications. Perelman, Grisha March 10, Ricci flow with surgery on three-manifolds. Perelman, Grisha July 17, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.

Bruce Kleiner, John Lott.

### Advanced Studies in Pure Mathematics

John W. Morgan, Gang Tian. It's very obsolete , and does not contain much on short-term existence of solutions of Ricci flow. You might try Terence Tao's blog notes from his course on Perelman's proof. He assumes a basic understanding of Riemannian geometry or at least goes over the requisite bits of it only very quickly so you may also want to start with a book on Riemannian geometry Tao himself was using Peter Petersen's book. It is aimed at explaining Ricci flow with surgery or rather a variation called Ricci flow with bubbling-off and the proof of geometrization to topologists and geometers, and the analysis of Ricci flow is mostly used as a blackbox, so that may suit you or not. In my view, the best place to start learning about Ricci flow is Hamilton's famous paper "Three-manifolds with positive Ricci curvature," modulo the short-time existence section.

DeTurck later came up with an easier way to prove short-time existence of solutions. I like Hamilton's paper because it introduces the reader to the intense tensor computations involved in Ricci flow theory and requires only basic Riemannian geometry: Riemannian metrics, the Levi-Civita connection, covariant differentiation of tensor fields, parallel transport, geodesics, the exponential map, normal coordinates, curvature, the Hopf-Rinow theorem, variations of energy and Myers' theorem come to mind.

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Moreover, Hamilton proves the tensor maximum principle and illustrates the power of maximum principle techniques. From there, you should be equipped to handle expository work on the Ricci flow. All of the sources mentioned above are great; I particularly like Simon Brendle's book "Ricci Flow and the Sphere Theorem" as a reference for convergence theory. It develops Hamilton's Ricci flow from the ground up leading to Brendle and Schoen's proof of the differentiable sphere theorem and also provides a very good overview of the required geometry in the first chapter.

This normalized equation preserves the volume of the metric. However, the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed, then the Ricci flow would usually only be defined for small negative times. This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time. Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

Ricci solitons are Ricci flows that may change their size but not their shape up to diffeomorphisms. The Ricci flow was utilized by Richard S.

General relativity - Ricci flow physics

Hamilton to gain insight into the geometrization conjecture of William Thurston , which concerns the topological classification of three-dimensional smooth manifolds. Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston; the possibilities, called Thurston model geometries , include the three-sphere S 3 , three-dimensional Euclidean space E 3 , three-dimensional hyperbolic space H 3 , which are homogeneous and isotropic , and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic.

This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes. Hamilton's idea was that these special metrics should behave like fixed points of the Ricci flow, and that if, for a given manifold, globally only one Thurston geometry was admissible, this might even act like an attractor under the flow.

Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time. This doesn't prove the full geometrization conjecture, because the most difficult case turns out to concern manifolds with negative Ricci curvature and more specifically those with negative sectional curvature.

Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem , the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

Note that the term "uniformization" suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" suggests placing a geometry on a smooth manifold. Geometry is being used here in a precise manner akin to Klein 's notion of geometry see Geometrization conjecture for further details. In particular, the result of geometrization may be a geometry that is not isotropic.

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In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

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## Ricci Flow and the Poincaré Conjecture

The Ricci flow does not preserve volume, so to be more careful, in applying the Ricci flow to uniformization and geometrization one needs to normalize the Ricci flow to obtain a flow which preserves volume. If one fails to do this, the problem is that for example instead of evolving a given three-dimensional manifold into one of Thurston's canonical forms, we might just shrink its size. It is possible to construct a kind of moduli space of n-dimensional Riemannian manifolds, and then the Ricci flow really does give a geometric flow in the intuitive sense of particles flowing along flowlines in this moduli space.

Hamilton showed that a compact Riemannian manifold always admits a short-time Ricci flow solution. Later Shi generalized the short-time existence result to complete manifolds of bounded curvature.