The Einstein-Hilbert functional and the Sasaki-Futaki invariant

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 [7] a polarized Sasaki manifold is K-semistable if only if it has constant scalar curvature.