Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries

Abstract

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □gϕ+σϕ = (ϕ,∂ϕ) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary

PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

“…The Klein-Gordon equation on asymptotically anti-de Sitter (aAdS) spacetimes was studied in a number of works in the last several years. We refer the reader to [YG,Gal,Bac,Hol,War1,HS3,EK,HW,HS1,HS2] to mention only a few. Notably, the results include well-posedness for the Klein-Gordon equation by Vasy [Vas4] and Holzegel [Hol] in the case of Dirichlet boundary conditions, as well as well-posedness for Neumann and Robin boundary conditions by Warnick [War1].…”

Killing boundary data for anti-de Sitter-like spacetimes

“…An alternative approach to the analysis of continuous symmetries in anti-de Sitter-like spacetimes has been started in [9,10]. In this work, the objective is to encode the existence of a Killing vector solely through conditions on the conformal boundary -in the spirit of the principle of holography.…”

A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes

“…A weaker variant of the unique continuation property was established recently by Holzegel and Shao [HS16,HS17], for more restrictive classes of solutions of P u = 0. For convenience our terminology refers to Dirichlet boundary conditions and linear fields exclusively.…”