Long time existence of Minimizing Movement solutions of Calabi flow

TL;DR

This paper demonstrates the long-term existence of minimizing movement solutions for the Calabi flow using DeGiorgi's framework, and shows that regular solutions are smooth, advancing understanding of geometric flow behavior.

Contribution

It introduces a new approach to Calabi flow via minimizing movements and proves long-time existence and regularity results for these solutions.

Findings

Long time existence of minimizing movement solutions for Calabi flow.

Regular minimizing movements are smooth solutions.

Establishment of a priori regularity for solutions.

Abstract

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth solutions to Calabi flow.