Some comments on gravitational entropy and the inverse mean curvature flow

The Geroch-Wald-Jang-Huisken-IImanen approach to the positive energy problem 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) x R, in terms of the cosmological constant. We also show how the method gives a lower bound 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.