Laplace comparison on Kähler Ricci flow and convergence

Abstract: We first prove a uniform integral Laplace comparison result for the K\"ahler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors, we give a direct and independent proof of the Hamilton-Tian conjecture on convergence of K\"ahler-Ricci flows, modulo a codimension 4 singular set. We also expounded on some existing literature on this conjecture.