On the Scalar Curvature of Einstein Manifolds

TL;DR

This paper demonstrates the existence of high-dimensional compact manifolds with pairs of Einstein metrics having opposite scalar curvatures, providing counterexamples to a longstanding conjecture.

Contribution

It constructs explicit counterexamples to a conjecture by showing certain manifolds admit Einstein metrics with opposite scalar curvatures.

Findings

Existence of manifolds with Einstein metrics of opposite signs

Counterexamples to Besse's conjecture

Use of Barlow surface deformations with ample canonical bundle

Abstract

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.