Differential Geometry Seminar: Higher-order estimates for collapsing Calabi-Yau metrics

Seminar | April 30 | 2:10-3 p.m. | 939 Evans Hall

 Hans-Joachim Hein, Fordham University

 Department of Mathematics

Consider a compact Calabi-Yau manifold \(X\) with a holomorphic vibration \(F: X \rightarrow B\) over some base \(B\), together with a "collapsing" path of Kahler classes of the form \([F^*\omega _B] + t [\omega _X]\) for \(t \in (0,1]\). Understanding the limiting behavior as \(t \rightarrow 0\) of the Ricci-flat Kahler forms representing these classes is a basic problem in geometric analysis that has attracted a lot of attention since the celebrated work of Gross-Wilson (2000) on elliptically fibered K3 surfaces. The limiting behavior of these Ricci-flat metrics is still not well-understood in general even away from the singular fibers of \(F\). A key difficulty arises from the fact that Yau's higher-order estimates for the complex Monge-Ampere equation heavily depend on bounds on the curvature tensor of a suitable background metric, but such bounds are simply not available in this collapsing situation. I will explain recent joint work with Valentino Tosatti where we manage to bypass Yau's method in some cases, proving higher-order estimates even though the background curvature blows up.