Heat kernel bounds, ancient $\kappa$ solutions and the Poincar\'e conjecture

TL;DR

This paper derives Gaussian bounds for heat kernels on ancient Ricci flow solutions, providing a new, shorter proof of Perelman's classification and a simplified proof of the Poincaré conjecture.

Contribution

It introduces Gaussian heat kernel bounds for ancient solutions and offers a novel, streamlined proof of the Poincaré conjecture avoiding reduced distance and volume.

Findings

Gaussian upper bounds for heat kernels on ancient solutions

A new proof of Perelman's classification of ancient solutions

A simplified proof of the Poincaré conjecture

Abstract

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.