We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose–type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi–Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.
"Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds." J. Differential Geom. 84 (1) 191 - 229, January 2010. https://doi.org/10.4310/jdg/1271271798