Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant

TL;DR

This work demonstrates that static vacuum spacetimes with negative cosmological constant ($\Lambda<0$) admit a rich family of geodesically complete, non-singular solutions with prescribed conformal infinity, in contrast to rigidity expectations. By developing an AH-Einstein filling framework and analyzing isometry extensions from the conformal boundary via a twist-form formalism, the authors show that boundary globally static actions can force interior staticity and yield explicit warped-product metrics. They prove existence and, in symmetric cases, uniqueness results for AdS fillings including AdS-Schwarzschild and Horowitz-Myers solitons across sphere, torus, and higher-genus boundaries, while also establishing geodesic completeness and global hyperbolicity for the resulting Lorentzian spacetimes. The findings significantly broaden the landscape of static AdS vacua, linking conformal infinity data to a large, structured set of fillings and clarifying the role of topology and boundary data in determining interior geometry. Practical impact lies in understanding the rigidity (or lack thereof) of negative-$\Lambda$ spacetimes and in providing concrete families of globally static AdS solutions for theoretical and mathematical physics applications.

Abstract

We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established.