Unique continuation from infinity in asympotically Anti-de Sitter spacetimes II: Non-static boundaries

TL;DR

This paper extends unique continuation results for wave equations on asymptotically Anti-de Sitter spacetimes to include non-static boundary metrics, using new Carleman estimates and geometric foliations.

Contribution

It introduces generalized Carleman estimates and a geometric foliation approach for non-static boundary metrics in aAdS spacetimes, advancing unique continuation theory.

Findings

Established new Carleman estimates for non-static boundary metrics.

Proved unique continuation results for linear and nonlinear wave equations.

Laid groundwork for future analysis of Einstein equations in aAdS spacetimes.

Abstract

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations $\Box_g \phi + \sigma \phi = \mathcal{G} ( \phi, \partial \phi )$ on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting non-static 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 pseudoconvex hypersurfaces near the conformal boundary.