On irregular Sasaki-Einstein metrics in dimension 5

Sasakian geometry can be seen as the odd-dimensional counterpart of Kaehler geometry. Indeed, a /regular/ Sasakian manifold M is a circle bundle over some Kaehler manifold Z. In this situation the Sasakian geometry of M and the Kaehler geometry of Z are closely related to each other. For example the problem of finding a Sasaki-Einstein metric on M is equivalent to the problem of finding a Kaehler-Einstein metric on Z. However, in the so-called /irregular/ case this approach breaks down. On the other hand, one also obtains a new tool in this situation: a torus action of higher rank. In this talk I will explain how to make use of this new tool in order to prove the the existence or non-existence of irregular Sasaki-Einstein metrics on certain 5-manifolds.