Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces

Abstract

In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.

References in corpus (11)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• Effective versions of the positive mass theorem
• Mass rigidity for hyperbolic manifolds
• Positive scalar curvature with skeleton singularities
• Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow
• Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions
• The hyperbolic positive energy theorem
• Scalar curvature rigidity of geodesic balls in S^n
• The dihedral rigidity conjecture for n-prisms
• The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times
• Measuring Mass via Coordinate Cubes

Cited by in corpus (2)

• Dihedral rigidity for cubic initial data sets
• Area, Scalar Curvature, and Hyperbolic 3-Manifolds