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.

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关于多封闭流的 Calabi 型估计

多封闭流的规律性理论取决于假设与背景度量一致等价，获得度量的 $C^{\alpha}$ 规律性。这个估计是在 \cite{StreetPCFBI} 中通过改编自 Evans-Krylov 的思想而建立的，关键输入是由 $T \oplus T^*$ 上定义的相关“广义度量”满足的尖锐微分不等式。在这项工作中，我们通过简化的证明给出了这种估计的锐化形式。首先，我们展示了广义度量本身通过自然曲率量演化，这会很快导致对关联陈连接的估计，类似于并概括了 Calabi-Yau 对复杂 Monge Ampere 方程的 $C^3$ 估计。