  We show the existence of isoperimetric regions of sufficiently large volumes
in general asymptotically hyperbolic three manifolds. Furthermore, we show that
large coordinate spheres in compact perturbations of
Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the
context of the asymptotically hyperbolic Penrose inequality.
  Our results require that the scalar curvature of the metric satisfies
$R_{g}\geq -6$, and we construct an example of a compact perturbation of
Schwarzschild-anti-deSitter without $R_{g}\geq -6$ so that large centered
coordinate spheres are not isoperimetric. The necessity of scalar curvature
bounds is in contrast with the analogous uniqueness result proven by Bray for
compact perturbations of Schwarzschild, where no such scalar curvature
assumption is required.
  This demonstrates that from the point of view of the isoperimetric problem,
mass behaves quite differently in the asymptotically hyperbolic setting
compared to the asymptotically flat setting. In particular, in the
asymptotically hyperbolic setting, there is an additional quantity, the
"renormalized volume," which has a strong effect on the large-scale geometry of
volume.
