A note on the uniqueness of solutions for the Yamabe problem

TL;DR

This paper establishes the local uniqueness of solutions to the Yamabe problem near Einstein metrics, extending Obata's criterion to a broader class of conformal metrics.

Contribution

It proves the uniqueness of Yamabe solutions in conformal classes close to Einstein metrics, excluding the standard sphere, thus generalizing Obata's criterion.

Findings

Uniqueness of Yamabe solutions near Einstein metrics

Extension of Obata's uniqueness criterion

Applicability to conformal classes close to Einstein metrics

Abstract

We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.