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The rigidity of the Riemannian positive mass theorem for asymptoticallyhyperbolic manifolds states that the total mass of such a manifold is zero ifand only if the manifold is isometric to the hyperbolic space. This leads tostudy the stability of this statement, that is, if the total mass of anasymptotically hyperbolic manifold is almost zero, is this manifold close tothe hyperbolic space in any way? Motivated by the work of Huang, Lee andSormani for asymptotically flat graphical manifolds with respect to intrinsicflat distance, we show the intrinsic flat stability of the positive masstheorem for a class of asymptotically hyperbolic graphical manifolds byadapting the positive answer to this question provided by Huang, Lee and thethird named author.

Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds