A Gauge-Invariant Unique Continuation Criterion for Waves in Asymptotically Anti-de Sitter Spacetimes

Abstract

We reconsider the unique continuation property for a general class of tensorial Klein–Gordon equations of the form □gϕ+σϕ=G(ϕ,∇ϕ),σ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Box _{g} \\phi + \\sigma \\phi = {\\mathcal {G}}(\\phi ,\ abla \\phi ) \ ext {,} \\qquad \\sigma \\in {\\mathbb {R}} \\end{aligned}$$\\end{document}on a large class of asymptotically anti-de-Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author (Holzegel and Shao in Commun Math Phys 347(3):723–775, 2016; Commun Partial Differ Equ 42(12):1871–1922, 2017; McGill and Shao in Class Quantum Gravity 38(5):054001, 2021) (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: We replace the so-called null convexity criterion—the key geometric assumption on the conformal boundary needed in McGill and Shao (2021) to establish the unique continuation properties—by a more general criterion that is also gauge invariant.Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary.Similar to McGill and Shao (2021), we connect the failure of our generalized criterion to the existence of certain null geodesics near the conformal boundary. These geodesics are closely related to the classical Alinhac-Baouendi counterexamples to unique continuation (Alinhac and Baouendi in Math Z 220(4):561–568, 1995). Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author (Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes (in preparation), 2022).