Einstein and scalar flat Riemannian metrics

Abstract

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critical point equation if, and only if, it is either Einstein, or scalar flat.

arXiv commentModified the proof of main Theorem (4) to simplify and provide details. Use now a pseudodifferential pertubation of the Yamabe flow instead of the Ricci flow, and a simpler and of lesser scope Lemma 2. The solution of the perturbed Yamabe flow here defined then leads to the desired contradiction when we assume that a critical metric of the functional does not have constant scalar curvature