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

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

2008-12-12

arxiv.org/abs/0812.2460v2

Mathematics

Differential Geometry

more references added, especially [CL]; some details added on p12-14

Scientific paper

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $\kappa$ 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 Poincar\'e conjecture without using reduced distance and reduced volume.

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