Rigidity of non-compact static domains in hyperbolic space via positive mass theorems

TL;DR

This paper establishes a positive mass theorem for certain non-compact static domains in hyperbolic space, revealing rigidity properties and extending previous results on boundary deformations under energy conditions.

Contribution

It introduces a new notion of staticity for non-compact domains in hyperbolic space and proves a positive mass theorem with rigidity, using elliptic boundary conditions on spinors.

Findings

Positive mass theorem for non-compact static hyperbolic domains

Rigidity result preventing certain boundary deformations

Extension of results by Souam on mean curvature bounds

Abstract

We single out a notion of staticity which applies to any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For (time-symmetric) initial data sets modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both in the interior and along the boundary) whose rigidity statement retrieves, among other things, a sharper version of a recent result by Souam to the effect that no such hypersurface admits a compactly supported deformation keeping the original lower bound on the mean curvature. A key ingredient in our approach is the consideration of a family of elliptic boundary conditions on spinors interpolating between chirality and MIT bag boundary conditions.