Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective

Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.

Hamilton–Perelman 三维庞加莱猜想证明的 Lean 形式化

A Lean Formalization of the Hamilton--Perelman Proof of the Three-Dimensional Poincaré Conjecture

本文用 Lean 形式化证明了三维庞加莱猜想，结合 Moise 定理，采用 Ricci 流与手术及有限时间消没的 Hamilton–Perelman 方法。

Ricci flow on Poincare' and Thurston's Geometrization Conjecture

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

Ricci Flow and the Poincare Conjecture

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.

Another way of answering Henri Poincare's fundamental question

After G. Perelman's solution of the Poincare Conjecture, this is a different way toward it. Given a simply connected, closed 3-manifold M, we produce a homotopy disc H, which arises from M by a finite sequence of simple modifications and, almost miraculously, can be imbedded into the ordinary space R^3. It follows that H is a disc, hence M is a sphere. In order to construct H, we use a special stratification of M, based on the fact that M is simply connected.

Heat kernel bounds, ancient $κ$ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $κ$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \cite{ChZ:1} lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.