Higher-order estimates for collapsing Calabi-Yau metrics (Q827531)

The authors study Ricci-flat Kähler metrics on compact Calabi-Yau manifolds. The main result proves a general \(C^{\alpha}\) estimate for collapsing Calabi-Yau metrics on the total spaces of proper holomorphic submersions over the unit ball in \(\mathbb C^n\). The proofs presented in the paper require new methods, the usual ones used among others by Calabi, Evans-Krylov, or Caffarelli do not work here. What allows the authors to go beyond known estimates is the use of iterated blowup arguments and linear and non-linear Liouville theorems in cylinders. The arguments are \textit{local on the base} but global on the fibers. If the fibers of the proper submersion are pairwise biholomorphic, the methods provide a uniform \(C^\infty\) estimate.