The Near-Boundary Geometry of Einstein-Vacuum Asymptotically Anti-de Sitter Spacetimes

TL;DR

This paper investigates the near-boundary geometry of vacuum asymptotically Anti-de Sitter spacetimes with finite regularity and arbitrary boundary topology, deriving limits and constructing expansions to facilitate future symmetry and correspondence theorems.

Contribution

It introduces a method to analyze the boundary behavior of such spacetimes, allowing for arbitrary boundary topology and finite regularity, and constructs partial Fefferman--Graham expansions.

Findings

Derived limits of geometric quantities at the boundary.

Constructed partial Fefferman--Graham expansions.

Established groundwork for symmetry and gravity--boundary theorems.

Abstract

We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman--Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity--boundary correspondence theorems for vacuum asymptotically Anti-de Sitter spacetimes.