Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 099, 8 pages      arXiv:2007.12563
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

Chao Li
Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA

Received July 27, 2020, in final form September 30, 2020; Published online October 06, 2020

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.

Key words: dihedral rigidity; scalar curvature; comparison theorem; hyperbolic manifolds.

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