Higher-order estimates for collapsing Calabi-Yau metrics

TL;DR

This paper establishes uniform regularity estimates for collapsing Calabi-Yau metrics, overcoming degenerating background geometry by using blowup techniques and Liouville theorems, with applications to compact Calabi-Yau manifolds.

Contribution

It introduces new methods for obtaining regularity estimates in collapsing Calabi-Yau metrics, including sharp Schauder estimates on cylinders and conditions for uniform C^infinity bounds.

Findings

Proves a uniform C^alpha estimate for collapsing Calabi-Yau metrics.

Develops sharp Schauder estimates for the Laplacian on cylinders.

Achieves uniform C^infinity estimates when fibers are biholomorphic.

Abstract

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.