On the K\"ahler-Ricci flow near a K\"ahler-Einstein metric

TL;DR

This paper proves that on a Fano manifold, the Kahler-Ricci flow starting close to a Kahler-Einstein metric converges polynomially to it, using Perelman's unctional and a Lojasiewicz inequality.

Contribution

It establishes polynomial convergence of the Kahler-Ricci flow near a Kahler-Einstein metric on Fano manifolds through a novel gauge modification and functional analysis.

Findings

Flow converges polynomially to Kahler-Einstein metric

Convergence depends on initial proximity to the Einstein metric

Uses Lojasiewicz inequality for Perelman's unctional

Abstract

On a Fano manifold, we prove that the Kahler-Ricci flow starting from a Kahler metric in the anti-canonical class which is sufficiently close to a Kahler-Einstein metric must converge in a polynomial rate to a Kahler-Einstein metric. The convergence can not happen in general if we study the flow on the level of Kahler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's \mu functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's \mu functional near a critical point.