We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which Arnowitt-Deser-Misner (ADM) mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n ≥ 3 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstant time evolution of the asymptotically hyperbolic mass is natural in light of a conjecture of Horowitz and Myers, and is a test of that conjecture. Finally, we use a simple parabolic scaling argument to produce a heuristic "derivation" of the constancy of ADM mass under asymptotically flat Ricci flow, starting from our decay formula for the asymptotically hyperbolic mass under the curvature-normalized flow.
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