In a space-time with cosmological constant $\Lambda >0\mathrm{}$ and matter satisfying the dominant energy condition, the area of a black or white hole cannot exceed $\frac{4\pi }{\Lambda }$. This applies to event horizons where defined, i.e., in an asymptotically de Sitter space-time, and to outer trapping horizons (cf. apparent horizons) in any space-time. The bound is attained if and only if the horizon is identical to that of the degenerate "Schwarzschild-de Sitter" solution. This yields a topological restriction on the event horizon, namely that components whose total area exceeds $\frac{4\pi }{\Lambda }$ cannot merge. We discuss the conjectured isoperimetric inequality and implications for the cosmic censorship conjecture.

• Received 30 August 1993

DOI:https://doi.org/10.1103/PhysRevD.49.5080