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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.
53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
35K55 Nonlinear parabolic equations
Full Text: DOI
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