Gromov-Hausdorff limits of immortal K\"ahler-Ricci flows

TL;DR

This paper proves that the normalized K"ahler-Ricci flow on certain compact K"ahler manifolds converges to a metric completion of a twisted K"ahler-Einstein metric, confirming a conjecture in the minimal model program.

Contribution

It establishes the Gromov-Hausdorff convergence of the normalized K"ahler-Ricci flow to the canonical model's metric completion, advancing understanding of geometric flows in algebraic geometry.

Findings

Flow converges to the metric completion of the twisted K"ahler-Einstein metric.

Confirms Song-Tian's conjecture on the flow's limit behavior.

Provides a link between Ricci flow and minimal model program.

Abstract

We show that the normalized K\"ahler-Ricci flow on a compact K\"ahler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted K\"ahler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.