We prove a uniform $C^\alpha$ estimate for collapsing Calabi–Yau metrics on the total space of a proper holomorphic submersion over the unit ball in $\mathbb{C}^m$. The usual methods of Calabi, Evans–Krylov, Caffarelli, et al. do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform $C^\infty$ estimate. We then apply these local results to the case of collapsing Calabi–Yau metrics on compact Calabi–Yau manifolds. In this global setting, the $C^0$ estimate required as a hypothesis in our new local $C^\alpha$ and $C^\infty$ estimates is known to hold thanks to earlier work of the second-named author.

中文翻译：

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崩溃的Calabi–Yau指标的高阶估计

我们证明了对于在$ \ mathbb {C} ^ m $中单位球上适当的全纯浸入式总浸没空间上的Calabi–Yau度量值崩溃，统一的$ C ^ \ alpha $估计。Calabi，Evans-Krylov，Caffarelli等的常用方法。不适用于此设置，因为背景几何会退化。相反，我们依赖于爆破参数以及圆柱上的线性和非线性Liouville定理。特别地，作为中间步骤，我们使用这样的论据证明了圆柱上拉普拉斯算子的新的Schauder估计值。如果浸没的纤维是成对的双全形的，则我们的方法将得出统一的$ C ^ \ infty $估计值。然后，我们将这些局部结果应用于在紧凑的Calabi-Yau流形上折叠Calabi-Yau度量的情况。在这种全局设置下，由于第二作者的较早工作，已知在新的本地$ C ^ \ alpha $和$ C ^ \ infty $估计中作为假设所需的$ C ^ 0 $估计成立。