Higher-order estimates for collapsing Calabi–Yau metrics
Higher-order estimates for collapsing Calabi–Yau metrics
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DOI：
10.4310/cjm.2020.v8.n4.a1
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
H. Hein;Valentino Tosatti
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文献类型：
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作者：
H. Hein;Valentino Tosatti
证明了单位球上的真全纯浸入的全空间上的塌缩Calabi-Yau度量的一致C^alpha估计。Calabi、Evans-Krylov和Caffarelli的常用方法不适用于此设置，因为背景几何体会退化。相反，我们依靠爆破参数和线性和非线性刘维尔定理的气缸。特别是，作为一个中间步骤，我们使用这样的参数来证明尖锐的新的Schauder估计的拉普拉斯圆柱。如果浸没的纤维是成对双全纯的，我们的方法产生一致的C^infinity估计。然后，我们将这些局部结果的情况下，崩溃的卡-丘度量紧卡-丘流形。在这个全局设置中，由于第二位作者的早期工作，我们的新的局部C^alpha和C^infinity估计中作为假设所需的C^0估计已知是成立的。
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 C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli 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^infinity 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^infinity estimates is known to hold thanks to earlier work of the second-named author.