Some Comments on Gravitational Entropy and the Inverse Mean Curvature Flow

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 $\Sigma_g \times {\Bbb R}$, 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.