Quasi-local mass functionals and generalized inverse mean curvature flow

Let M be an asymptotically flat 3-manifold with non-negative scalar curvature. In H. Bray, A family of quasi-local mass functionals with monotone flows, Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003, Hubert Bray defines a family of quasi-local mass functionals which are monotone for surfaces smoothly satisfying a certain generalization of inverse mean curvature flow in M. We show that a weak solution in the sense of Huisken-Ilmanen exists for a wide class of flows including these with monotone quasi-local mass functionals, and we show that the monotonicity holds for the weak flow as well. As shown in H. Bray, A family of quasi-local mass functionals with monotone flows, Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003, a Penrose-type inequality for connected surfaces is an immediate corollary.

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Published 1 January 2008