Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long‐standing problem of uniform diameter bounds and Gromov–Hausdorff convergence of the Kähler–Ricci flow, for both finite‐time and long‐time solutions.

中文翻译：

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卡勒几何中的直径估计

建立了凯勒度量的直径估计，该估计仅需要熵界并且不需要里奇曲率的下界。该证明建立在最近用于估计 Monge-Ampère 方程的偏微分方程技术的基础上，并进行了一项关键改进，允许余维数的体积形式的简并严格大于 1。因此，我们解决了长期存在的均匀直径边界问题和 Kähler-Ricci 流的 Gromov-Hausdorff 收敛问题，无论是有限时间解还是长期解。