The wave equation on asymptotically anti de Sitter spaces

Abstract

In this paper we describe the behavior of solutions of the Klein–Gordon equation, $\left({\square }_g+\lambda \right)u=f$, on Lorentzian manifolds $\left(X^{\circ },g\right)$ that 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 $ĝ$ on $X$, where $X$ has a nontrivial boundary, in the sense that $g=x^{-2}ĝ$, with $x$ a boundary defining function. The boundary is conformally timelike for these spaces, unlike asymptotically de Sitter spaces studied before by Vasy and Baskin, which are similar but with the boundary being conformally spacelike.

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, ${\left(n-1\right)}^2∕4$. These have been known before under additional assumptions. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at $\partial X$) of regular solutions. We also define the scattering operator, which in this case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is shown that below the Breitenlohner–Freedman bound, the Klein–Gordon equation behaves much like it would for the conformally related metric, $ĝ$, with Dirichlet boundary conditions, for which propagation of singularities was shown by Melrose, Sjöstrand and Taylor, though the precise form of the asymptotics is different.