Regularity of K\"ahler-Ricci flow

TL;DR

This paper establishes a regularity theorem for the K"ahler-Ricci flow on Fano manifolds, with implications for the flow's limiting behavior and an extension of partial $C^0$ estimates under regularity assumptions.

Contribution

It introduces a new regularity theorem for K"ahler-Ricci flow on Fano manifolds and applies it to analyze the flow's limits and extend partial $C^0$ estimates.

Findings

Regularity theorem for K"ahler-Ricci flow on Fano manifolds.

Application to the limiting behavior of the flow on Fano 3-manifolds.

Extension of partial $C^0$ estimates under regularity assumptions.

Abstract

In this short note we announce a regularity theorem for K\"ahler-Ricci flow on a compact Fano manifold (K\"ahler manifold with positive first Chern class) and its application to the limiting behavior of K\"ahler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate to the K\"ahler-Ricci flow under the regularity assumption, which extends previous works on K\"ahler-Einstein metrics and shrinking K\"ahler-Ricci solitons (cf. \cite{Ti90}, \cite{DoSu12}, \cite{Ti12}, \cite{PSS12}). The detailed proof will appear in \cite{TiZh13}.