Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds

Details Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds

2007-11-27

arxiv.org/abs/0711.4335v1

Mathematics

Differential Geometry

35 pages, submitted

Scientific paper

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 $S^2$.

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