On a Calabi-type estimate for pluriclosed flow

TL;DR

This paper improves the regularity estimate for pluriclosed flow by providing a sharper, simplified proof that relates the evolution of the generalized metric to curvature, extending classical estimates in complex geometry.

Contribution

It offers a refined estimate and a simplified proof for the regularity of pluriclosed flow, connecting the generalized metric evolution to curvature quantities.

Findings

Sharpened $C^{eta}$ regularity estimate for pluriclosed flow

Simplified proof based on curvature evolution of the generalized metric

Extension of Calabi-Yau's $C^3$ estimate to pluriclosed flow

Abstract

The regularity theory for pluriclosed flow hinges on obtaining $C^{\alpha}$ regularity for the metric assuming uniform equivalence to a background metric. This estimate was established in \cite{StreetsPCFBI} by an adaptation of ideas from Evans-Krylov, the key input being a sharp differential inequality satisfied by the associated `generalized metric' defined on $T \oplus T^*$. In this work we give a sharpened form of this estimate with a simplified proof. To begin we show that the generalized metric itself evolves by a natural curvature quantity, which leads quickly to an estimate on the associated Chern connections analogous to, and generalizing, Calabi-Yau's $C^3$ estimate for the complex Monge Ampere equation.