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On a Calabi-type estimate for pluriclosed flow. (English) Zbl 1436.53074
Summary: 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 $$T \oplus T^\ast$$. 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-Ampère equation.
##### MSC:
 53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows) 35K55 Nonlinear parabolic equations
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##### References:
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