Kahler-Einstein metrics and the Kahler-Ricci flow on log Fano varieties

We prove the existence and uniqueness of Kahler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kathler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kahler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.