Hamilton-Perelman's Proof of the Poincar\'e Conjecture and the Geometrization Conjecture

We discuss some of the key ideas of Perelman's proof of Poincar\'e's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.

We give a brief survey of Hamilton's program for 3-manifolds as an approach toward Thurston's Geometrization Conjectre.

This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS book style.

In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery...

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov sp...

In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate

Although the Poincare' and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow previously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan (Ann.Phys.163(1985)318-419). In his work the renormalization group flow for a nonlinear sigma model in 2+e dimensions was obtained and studied. For e=0, by approximating the beta function for...

So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincar\'e's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have...

This article reports recent developments of the research on Hamilton's Ricci flow and its applications.