# Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry

## TL;DR

This paper demonstrates that spherically symmetric asymptotically hyperbolic manifolds with vanishing mass converge to hyperbolic space in the intrinsic flat sense, highlighting an almost rigidity property of the positive mass theorem.

## Contribution

It establishes the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds using intrinsic flat convergence, specifically for spherically symmetric cases.

## Key findings

- Sequences with zero mass converge to hyperbolic space in the intrinsic flat sense.
- The positive mass theorem exhibits an almost rigidity property in the hyperbolic setting.
- Intrinsic flat distance effectively measures convergence in this geometric context.

## Abstract

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.