The Mabuchi metric and the K\"ahler-Ricci flow

TL;DR

This paper demonstrates that on Fano manifolds, the convergence of the K"ahler-Ricci flow to a K"ahler-Einstein metric can be established through the integrability of the Ricci potential's L^2 norm over positive time.

Contribution

It introduces a new criterion linking Ricci potential integrability to the convergence of the K"ahler-Ricci flow on Fano manifolds.

Findings

Convergence follows from Ricci potential L^2 integrability.

Establishes a connection between Mabuchi metric and flow convergence.

Provides conditions for K"ahler-Einstein metric existence.

Abstract

In this paper we show that on a Fano manifold the convergence of the K\"ahler-Ricci flow to a K\"ahler-Einstein metric follows from the integrability of the $L^2$ norm of the Ricci potential for positive time.