Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces
Chao Li
TL;DR
This paper proves dihedral rigidity for parabolic polyhedrons in hyperbolic spaces, extending Gromov's comparison theory and relating to the positive mass theorem for asymptotically hyperbolic manifolds.
Contribution
It establishes a new dihedral rigidity phenomenon for polyhedrons in hyperbolic spaces, extending existing geometric comparison theories.
Findings
Dihedral rigidity for parabolic polyhedrons in hyperbolic spaces
Extension of Gromov's comparison theory to negative scalar curvature
Localization of the positive mass theorem in hyperbolic manifolds
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.
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