Bounded area theorems for higher-genus black holes

By a simple modification of Hawking's well known topology theorems for blackhole horizons, we find lower bounds for the areas of smooth apparent horizons and smooth cross sections of stationary black hole event horizons of genus g > 1 in four dimensions. For a negatively curved Einstein space, the bound is 4 pi(g - 1)/(- l) where l is the cosmological constant of the spacetime. This is complementary to the known upper bound on the area of g = 0 black holes in de Sitter spacetime. It also emerges that g > 1 quite generally requires a mean negative energy density on the horizon. The bound is sharp; we show that it is saturated by certain extreme, asymptotically locally anti-de Sitter spacetimes. Our results generalize a recent result of Gibbons.