H. Bray

Title: Global inequalities

Lecture 1: Generalization of the Hawking Mass
Abstract: We observe that the Hawking mass and inverse mean curvature flow can be generalized to give additional estimates of the ADM mass from new non-decreasing quasi-local mass functionals. These new estimates are the best known explicit estimates for how much matter is in a given region.

Lecture 2: Proof of the Poincaré Conjecture for 3-Manifolds With Yamabe Invariant Greater Than RP^3
Abstract: We (joint with Andre Neves) use techniques from physics, namely inverse mean curvature flow as defined by Geroch-Jang-Wald-Huisken-Ilmanen, to prove that the Yamabe Invariant of RP^3 equals Y_2, the energy of the constant curvature metric on RP^3. Furthermore, we prove that any 3-manifold with Yamabe invariant Y greater than Y_2 is either S^3 or a connect sum with an S^2 bundle over S^1. The Poincare conjecture for Y > Y_2 follows as a corollary. Also, the first five prime 3-manifolds, ordered by their Yamabe invariants, are S^3, the two S^2 bundles over S^1, RP^3, and RP^2 x S^1. The authors began looking at this problem after Richard Schoen pointed out a possible connection between the Penrose Inequality in General Relativity and the Yamabe Invariant of RP^3.

Lecture 3: Black Holes, the Penrose Conjecture, and Quasi-Local Mass
Abstract: We advertise the importance of the Penrose Conjecture (still open) for a general slice of a spactime and observe several applications to defining how much matter is in a given region of a space-time. We argue that the quasi-local mass of a region should not be a number but instead should be an interval in the real numbers. We submit that this is because gravitational energy generally refuses to say precisely what its location is and so does not like to answer the question Are you in this region? We define the inner and outer mass functions (a la Bartnik) of each region (which define the intervals endpoints) and then the total inner mass and the total outer mass of the general space-like slice. Finally, we conjecture that the total inner mass and total outer mass are always equal under very general assumptions. This gives a definition of total mass in great generality.

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Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and H. Bray