Near-Boundary Asymptotics and Unique Continuation for the AdS--Einstein--Maxwell System

TL;DR

This paper analyzes the near-boundary behavior and unique continuation properties of solutions to the coupled AdS-Einstein-Maxwell system, extending previous results and characterizing boundary data for the system.

Contribution

It extends asymptotic analysis and unique continuation results to the Einstein-Maxwell system, providing new insights into boundary data characterization and solution uniqueness.

Findings

Characterization of holographic boundary data for the coupled system

Proof of local unique continuation from the boundary

Extension of asymptotic results to nonlinear Einstein-Maxwell equations

Abstract

In this article, we extend the results of both Shao and Holzegel-Shao to the AdS-Einstein-Maxwell system $({M}, g, F)$. We study the asymptotics of the metric $g$ and the Maxwell field $F$ near the conformal boundary ${I}$ for the fully nonlinear coupled system. Furthermore, we characterise the holographic (boundary) data used in the second part of this work. We also prove the local unique continuation property for solutions of the coupled Einstein equations from the conformal boundary. Specifically, the prescription of the coefficients $(\mathfrak{g}^{(0)}, \mathfrak{g}^{(n)})$ in the near-boundary expansion of $g$, along with the boundary data for the Maxwell fields $(\mathfrak{f}^{0}, \mathfrak{f}^{1})$, on a domain ${D} \subset {I}$ uniquely determines $(g, F)$ near ${D}$. The geometric conditions required for unique continuation are identical to those in the vacuum case, regardless of the presence of the Maxwell fields. This work is part of the author's thesis.