Ricci flow and the scalar curvature rigidity of Einstein manifolds

TL;DR

This paper reviews recent advances connecting linear and dynamical stability with scalar curvature rigidity in Einstein manifolds, covering various types including closed, open, and noncompact cases, and their relation to the positive mass theorem.

Contribution

It synthesizes recent results on stability and scalar curvature rigidity of Einstein manifolds across different classes and their links to the positive mass theorem.

Findings

Linear stability relates to dynamical stability in Einstein manifolds.

Scalar curvature rigidity holds for various classes of Einstein manifolds.

Connections to the positive mass theorem are established for these classes.

Abstract

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are asymptotically locally Euclidean and asymptotically hyperbolic. For these classes, the relation to the positive mass theorem will also be explained.