Einstein and scalar flat Riemannian metrics
Abstract
On a given closed connected manifold of dimension two, or greater, we consider the squared -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.
Cite
@article{arxiv.1911.02706,
title = {Einstein and scalar flat Riemannian metrics},
author = {Santiago R Simanca},
journal= {arXiv preprint arXiv:1911.02706},
year = {2020}
}
Comments
Modified 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