Some Comments on Gravitational Entropy and the Inverse Mean Curvature Flow

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The Geroch-Wald-Jang-Huisken-Ilmanen approach to the positive energy problem to may be extended to give a negative lower bound for the mass of asymptotically Anti-de-Sitter spacetimes containing horizons with exotic topologies having ends or infinities of the form Σg × R, in terms of the cosmological constant. We also show how the method gives a lower bound for for the mass of time-symmetric initial data sets for black holes with vectors and scalars in terms of the mass, |Z(Q,P)| of the double extreme black hole with the same charges. I also give a lower bound for the area of an apparent horizon, and hence a lower bound for the entropy in terms of the same function |Z(Q,P)|. This shows that the so-called attractor behaviour extends beyond the static spherically symmetric case. and underscores the general importance of the function |Z(Q,P)|. There are hints that higher dimensional generalizations may involve the Yamabe conjectures.