Vol. 5, No. 1, 2012

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
The wave equation on asymptotically anti de Sitter spaces

András Vasy

Vol. 5 (2012), No. 1, 81–144

In this paper we describe the behavior of solutions of the Klein–Gordon equation, (g + λ)u = f, on Lorentzian manifolds (X,g) 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 = x2ĝ, 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 λ below the Breitenlohner–Freedman bound, (n 1)24. These have been known before under additional assumptions. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at 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.

asymptotics, wave equation, anti de Sitter space, propagation of singularities
Mathematical Subject Classification 2000
Primary: 35L05, 58J45
Received: 23 December 2009
Revised: 11 October 2010
Accepted: 22 December 2010
Published: 25 June 2012

Proposed: Maciej Zworski
Seconded: Terence Tao, Steve Zelditch
András Vasy
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
United States