An improvement of regularity result for pseudo Calabi flow

TL;DR

This paper demonstrates that pseudo Calabi flow becomes immediately smooth if starting volume form is close to smooth, and establishes long-term existence when initial data is near a cscK metric or on Fano manifolds with bounded volume form.

Contribution

It provides new regularity and long-time existence results for pseudo Calabi flow under volume form closeness conditions.

Findings

Flow becomes smooth immediately for initial data close to smooth volume form.

Long-time existence of flow when initial data is near cscK metric.

Similar regularity results on Fano manifolds with volume form bounds.

Abstract

In this paper, we observe that if the initial data of pseudo Calabi flow has volume form $C^0$ close to a smooth one, then the flow is immediately smooth for $t>0$. As an application, we show that if the initial data has volume form $C^0$ close to that of a cscK metric, then the pseudo Calabi flow exists for $t\in (0,+\infty)$. We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to $c_1(M)$.