On a Calabi-type estimate for pluriclosed flow

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 [14] 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 circle plus 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. (C) 2020 Elsevier Inc. All rights reserved.