Uniqueness of de Sitter and Schwarzschild-de Sitter spacetimes

We give a simple proof of the uniqueness of de Sitter and Schwarzschild de Sitter spacetime without assuming extra conditions on the conformal boundary at infinity. Such spacetimes are the only solutions in the static class satisfying Einstein equations R-4(alpha beta) = Lambda g(alpha beta)(4), where the cosmological constant A is positive, under appropriate boundary conditions. In the absence of black holes, that is, when the event horizon has only one component the unique solution is de Sitter solution. In the presence of a black hole, we get Schwarzschild de Sitter spacetime. The problem has important relevance in differential geometry.