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 much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1], and Sesum (2006) [27], . The current paper or Chow and Lu (2004) [6], combined with Chen and Zhu (2006) [4], and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.