  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.
