  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 ADM mass is constant during Ricci flow, we show that the mass of an
asymptotically hyperbolic manifold of dimension n>2 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.
