Bulk-boundary correspondences and unique continuation in asymptotically Anti-de Sitter spacetimes

TL;DR

This paper reviews mathematical results establishing a rigorous connection between bulk and boundary data in asymptotically Anti-de Sitter spacetimes, focusing on unique continuation and Carleman estimates relevant to the AdS/CFT correspondence.

Contribution

It presents new unique continuation results for Einstein-vacuum equations in aAdS spacetimes and develops Carleman estimates that underpin the bulk-boundary correspondence.

Findings

Proved a unique continuation theorem for Einstein-vacuum equations in aAdS spacetimes.

Developed novel Carleman estimates for wave equations on aAdS backgrounds.

Established foundational results supporting the AdS/CFT correspondence in dynamical settings.

Abstract

This article surveys the research presented by the author at the MATRIX Institute workshop "Hyperbolic Differential Equations in Geometry and Physics" in April 2022. The work is centered about establishing rigorous mathematical statements toward the AdS/CFT correspondence in theoretical physics, in particular in dynamical settings. The contents are mainly based on the recent paper with G. Holzegel that proved a unique continuation result for the Einstein-vacuum equations from asymptotically Anti-de Sitter (aAdS) conformal boundaries. We also discuss some preceding results, in particular novel Carleman estimates for wave equations on aAdS spacetimes, which laid the foundations toward the main correspondence theorems.