- 6/06/2022: Asymptotically conical Calabi-Yau manifolds
- 5/25/2021: The renormalized volume of a 4-dimensional Ricci-flat ALE space
- 9/10/2018: Higher-order estimates for collapsing Calabi-Yau metrics
June 6, 2022
TITLE: Asymptotically conical Calabi-Yau manifolds
ABSTRACT: In this talk I will report on the recent conclusion of an old project aiming to classify all complete non-compact Calabi-Yau manifolds asymptotic to a given Calabi-Yau cone at infinity. Joint work with Ronan Conlon.
May 25, 2021
TITLE: The renormalized volume of a 4-dimensional Ricci-flat ALE space
ABSTRACT: I will briefly review the convergence theory for non-collapsed Einstein 4-manifolds developed by Anderson-Cheeger, Bando-Kasue-Nakajima and Tian around 1990. This was the main precursor for the more recent higher-dimensional theory of Cheeger-Colding-Naber. However, several difficult problems have remained open even in dimension 4. I will focus on the structure of the possible bubbles and bubble trees in the 4-dimensional theory. In particular, I will review Kronheimer’s classical work on gravitational instantons and explain a recent result of Biquard-H concerning the renormalized volume of a 4-dimensional Ricci-flat ALE space.
September 10, 2018
TITLE: Higher-order estimates for collapsing Calabi-Yau metrics
ABSTRACT: Consider a compact Calabi-Yau manifold X with a holomorphic fibration F: X to 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 to 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 depend on bounds on the curvature tensor of a suitable background metric that are 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.