The interior regularity of the Calabi flow on a toric surface

TL;DR

This paper establishes uniform interior regularity estimates for solutions of the Calabi flow on toric surfaces, based on local curvature and distance estimates, ensuring controlled behavior within the flow's existence interval.

Contribution

It provides the first interior regularity results for the Calabi flow on toric surfaces, linking local geometric estimates to flow regularity.

Findings

Uniform interior estimates for u(t) under the Calabi flow

Control of Riemann curvature and geodesic distance during flow

Extension of regularity results to toric surface setting

Abstract

Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.