Toric Sasaki-Einstein geometry
Akito Futaki

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.