On scalar curvature rigidity of Vacuum Static Spaces

TL;DR

This paper extends scalar curvature rigidity results to small domains in vacuum static spaces, including hyperbolic spaces, and establishes global rigidity for conformal metric deformations with positive lapse functions.

Contribution

It generalizes local and global scalar curvature rigidity results to broader classes of vacuum static spaces and domains, including hyperbolic spaces and conformal deformations.

Findings

Local scalar curvature rigidity in small domains of vacuum static spaces.

Global scalar curvature rigidity for conformal deformations with positive lapse functions.

Domains in hyperbolic spaces exhibit scalar curvature rigidity.

Abstract

In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [15].