The Einstein-Hilbert functional and the Sasaki-Futaki invariant

TL;DR

This paper demonstrates that the Einstein-Hilbert functional can identify when the Sasaki-Futaki invariant vanishes, providing insights into the existence of constant scalar curvature Sasakian metrics and linking K-semistability to scalar curvature conditions.

Contribution

It establishes a connection between the Einstein-Hilbert functional and the Sasaki-Futaki invariant, offering new obstructions and criteria for constant scalar curvature Sasakian metrics.

Findings

The Einstein-Hilbert functional detects vanishing Sasaki-Futaki invariant.

K-semistability implies vanishing Sasaki-Futaki invariant under certain conditions.

K-semistability is equivalent to constant scalar curvature in specific Sasaki join manifolds.

Abstract

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.