Scalar Curvature Rigidity for asymptotically locally hyperbolic manifolds

TL;DR

This paper proves rigidity results for asymptotically locally hyperbolic manifolds with scalar curvature bounds using spinor methods, showing that certain geometric conditions imply the manifold must be a standard model.

Contribution

It introduces a spinor-based approach to establish scalar curvature rigidity for asymptotically locally hyperbolic manifolds, including explicit formulas in four dimensions.

Findings

Mass vanishes for conformally compact Einstein manifolds with spherical boundary

Rigidity results hold under scalar curvature lower bounds

Explicit invariants are derived in 4D cases

Abstract

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the 4-dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.