Higher genus horizons as least area surfaces

By a trivial modification of Hawking's well-known topology theorem for black holes, one can obtain a lower bound of 4 pi(g-1)/-Lambda for the area of an apparent horizon of genus g in a vacuum spacetime with cosmological constant Lambda < 0. We discuss a derivation in the special case of an apparent horizon at a moment of time symmetry, so that the horizon is a minimal surface.