The wave equation on asymptotically Anti-de Sitter spaces

Understand

In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity.

• Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metric g^ on X, where C has a non-trivial boundary, in the sense that g=x^{-2}g^, with x a boundary defining function.
• The boundary is conformally time-like for these spaces, unlike asymptotically de Sitter spaces, which are similar but with the boundary being conformally space-like.
• Here we show local well-posedness for the Klein-Gordon equation, and also global well-posedness under global assumptions on the (null)bicharacteristic flow, for lambda below the Breitenlohner-Freedman bound, (n-1)^2/4.