Valentino Tosatti (Northwestern University)

Title: Higher order estimates for collapsing Ricci-flat metrics.

Abstract: I will discuss the problem of obtaining uniform C^k estimates for solutions of a degenerating family of complex Monge-Ampere equations where the ellipticity is degenerating along the fibers of a fibration. Geometrically, the solutions of this family of PDE give Ricci-flat Kahler metrics on a Calabi-Yau manifold with a holomorphic fibration onto a lower-dimensional space. I will describe new estimates that prove a uniform C^{2,alpha} bound in general, and C^k bounds for all k when the smooth fibers are all pairwise isomorphic. This is joint work with H.-J. Hein.