The consistency and convergence of K-energy minimizing movements

TL;DR

This paper proves that K-energy minimizing movements align with smooth Calabi flow solutions when they exist, and shows long-term convergence of these flows to minimizers of the K-energy functional in Kahler geometry.

Contribution

It establishes the equivalence of K-energy minimizing movements with Calabi flow solutions and demonstrates convergence to energy minimizers in Kahler metrics.

Findings

K-energy minimizing movements agree with Calabi flow when smooth solutions exist

Long-term solutions minimize both K-energy and Calabi energy

Solutions converge to minimizers in the metric completion of Kahler metrics

Abstract

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing movements, these results imply that long time solutions to Calabi flow converge in the weak distance topology to minimizers of the K-energy functional on the metric completion of the space of Kahler metrics, assuming one exists.