Subscribe to Events

Abstract: This talk focuses on the recent resolution of two well-known conjectures in the study of Ricci-flat four manifolds (joint with Song Sun).

(1) Any volume collapsed limit of unit-diameter hyperkaehler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kaehler metric on the 2-sphere, or the unit interval.

(2) Any complete hyperkaehler 4-manifold with quadratically integrable curvature must have one of the following asymptotic model geometries: ALE, ALF, ALG, ALH, ALG* and ALH*.

Combining with known results, we have obtained a rather complete picture of collapsing hyperkaehler metrics bounded curvature energy in real dimension four (Gromov-Hausdorff limits, rescaling limits, bubble tree structure, etc).