Local and global scalar curvature rigidity of Einstein manifolds

TL;DR

This paper investigates the conditions under which Einstein manifolds exhibit scalar curvature rigidity, providing characterizations and constructing specific metric perturbations that decrease mass in certain models.

Contribution

It offers new characterizations of scalar curvature rigidity for both open and closed Einstein manifolds and constructs explicit mass-decreasing perturbations for notable metrics.

Findings

Characterizations of scalar curvature rigidity for Einstein manifolds

Construction of mass-decreasing perturbations of Schwarzschild and Taub-Bolt metrics

Identification of conditions preventing non-trivial volume-preserving deformations

Abstract

An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformation of the metric which increase the scalar curvature. We give various characterizations of scalar curvature rigidity for open Einstein manifolds as well as for closed Einstein manifolds. As an application, we construct mass-decreasing perturbations of the Riemannian Schwarzschild metric and the Taub-Bolt metric.