Towards the classification of static vacuum spacetimes with negative cosmological constant

We present a systematic study of static solutions of the vacuum Einstein eq uations with negative cosmological constant which asymptotically approach t he generalized Kottler ("Schwarzschild-anti-de Sitter") solution, within (m ainly) a conformal framework. We show connectedness of conformal infinity f or appropriately regular such spacetimes. We give an explicit expression fo r the Hamiltonian mass of the (not necessarily static) metrics within the c lass considered; in the static case we show that they have a finite and wel l-defined Hawking mass. We prove inequalities relating the mass and the hor izon area of the (static) metrics considered to those of appropriate refere nce generalized Kottler metrics. Those inequalities yield an inequality whi ch is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler blac k holes if the generalized Penrose inequality can be established. (C) 2001 American Institute of Physics.