Higher-order estimates for collapsing Calabi-Yau metrics

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.

arXiv comment53 pages, small changes, final version to appear in the Cambridge Journal of Mathematics

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Cited by in corpus (11)

• Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow
• Smooth asymptotics for collapsing Calabi-Yau metrics
• A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry
• Collapsing Calabi-Yau fibrations and uniform diameter bounds
• Twisted Kähler-Einstein Metrics and Collapsing
• Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces
• Collapsing immortal Kähler-Ricci flows
• On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
• Semipositive line bundles and (1,1)-classes
• Ricci-flat metrics on Calabi-Yau manifolds
• Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps