Abstract: 

This talk focuses on the recent resolution of the following three 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 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval.

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

(3) Any gravitational instanton can be compactified to an open dense subset of certain compact algebraic surface.
With the above classification results, we obtain a rather complete picture of the collapsing geometry of hyperkaehler four manifolds, i.e., classifications of Gromov-Hausdorff limit geometries, tangent cones, singularity formations, as well as all the rescaling bubbles.

Speaker：

Ruobing Zhang

Ruobing Zhang is currently an assistant professor at Princeton University. He obtained his Ph.D. at Princeton University in 2016 under the supervision of Alice Chang and Paul Yang. His research focuses on differential geometry and geometric analysis.

Zoom:

https://us02web.zoom.us/j/84187721220?pwd=b0h3V0RLZnVmSWRCS0ErcDJkNFYrUT09

Passwords: 430726