The Einstein–Hilbert Functional and the Sasaki–Futaki Invariant
Résumé
We show that the Einstein–Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki–Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki manifold has vanishing Sasaki–Futaki invariant. We then apply this result to show that under the right conditions on the Sasaki join manifolds of [6], a polarized Sasaki manifold admits constant scalar curvature Sasaki metric if it is K-polystable.