Toric Sasaki-Einstein geometry
A Sasakian manifold is an odd dimensional Riemannian manifold whose Riemannian cone is a Kähler
manifold. A Sasakian manifold inherits a contact structure, and its Reeb vector field generates
a flow which has transverse Kähler structure. Here, by transverse Kähler structure, we mean
a compatible collection of Kähler structures on local orbit spaces of the Reeb flow. Sasaki-Einstein manifolds
have been extensively studied by mathematicians and physicists in recent years.
A Sasakian manifold has an Einstein metric if and only if its Kähler cone is Ricci-flat,
and also if and only if the local orbit spaces of the Reeb flow have a transverse positive Kähler-Einstein
structure. In this talk I will focus on toric Sasaki-Einstein manifolds and associated toric Ricci-flat Kähler cones.
Ricci flow, mean curvature flow and their self-similar solutions are studied on
the associated Kähler cones or their resolutions.