K\"ahler-Einstein metrics and the K\"ahler-Ricci flow on log Fano varieties

TL;DR

This paper establishes the existence and uniqueness of Kähler-Einstein metrics on log Fano varieties with singularities, and analyzes the convergence of the Kähler-Ricci flow and Ricci iteration, improving previous results especially for smooth Fano manifolds.

Contribution

It proves new existence and convergence results for Kähler-Einstein metrics on singular Fano varieties and enhances understanding of Ricci flow and iteration without extra conditions.

Findings

Existence and uniqueness of Kähler-Einstein metrics on log Fano varieties with proper Mabuchi functional.

Smooth convergence of Ricci iteration on non-singular Fano manifolds without additional assumptions.

Weak convergence of the Kähler-Ricci flow independent of Perelman's estimates.

Abstract

We prove the existence and uniqueness of K\"ahler-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 K\"ahler-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 K\"ahler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.