Ricci flow and the metric completion of the space of Kahler metrics

TL;DR

This paper investigates the geometry of the space of Kähler metrics using Ricci flow, establishing the metric completion, and providing new stability criteria for Kähler-Einstein metrics on Fano manifolds.

Contribution

It characterizes the metric completion of the space of Kähler metrics and links Ricci flow convergence to metric stability criteria for Kähler-Einstein metrics.

Findings

Intrinsic and extrinsic distances are equivalent in the space of Kähler metrics.

The metric completion of the space of Kähler metrics is determined.

A new stability criterion for Kähler-Einstein metrics based on Ricci flow is established.

Abstract

We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kahler metrics, making contact with recent generalizations of the Calabi-Yau Theorem due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new analytic stability criterion for the existence of a Kahler-Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kahler-Ricci flow converges as soon as it converges in the metric sense.