Asymptotically hyperbolic $3$-manifolds show up naturally in the context of general relativity. I will show that, provided the mass is positive, these manifolds admit a unique foliation by stable spheres with constant mean curvature. In analogy with asymptotically flat manifolds, we can also formulate a Penrose inequality and I will explain, contrarlily to what was expected, why inverse mean curvature flow does not give a proof. This is joint work with Tian and the talk will be self contained.

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