Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds
Abstract
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 $S^2$.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2007
- DOI:
- 10.48550/arXiv.0711.4335
- arXiv:
- arXiv:0711.4335
- Bibcode:
- 2007arXiv0711.4335N
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- 53C44
- E-Print:
- 35 pages, submitted