The Kahler-Ricci flow and K-stability

TL;DR

This paper investigates conditions under which the Kähler-Ricci flow on Fano manifolds converges to a Kähler-Einstein metric, linking curvature bounds, energy functionals, and K-stability.

Contribution

It establishes that bounded curvature, K-polystability, and energy bounds imply the existence of Kähler-Einstein metrics on Fano manifolds.

Findings

Bounded curvature implies Mabuchi energy is bounded below.

K-polystability combined with curvature bounds leads to Kähler-Einstein metrics.

Small perturbations of cscK manifolds can admit cscK metrics if K-polystable.

Abstract

We consider the K\"ahler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a K\"ahler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.