Heat kernel bounds, ancient $κ$ solutions and the Poincaré conjecture
Abstract
Description
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.
more references added, especially [CL]; some details added on p12-14
more references added, especially [CL]; some details added on p12-14