V-Static Metrics and the Volume-Renormalised Mass

V-static metrics generalise the notion of static metrics, and stem from the work of Miao and Tam (Calc Var Partial Differ Equ 36(2):141–171, 2009), and Corvino, Eichmair, and Miao (Math Ann 357(2):551–584, 2013) on critical points of the volume functional over the space of compact manifolds with constant scalar curvature. In this article we show that these V-static metrics arise naturally in the context of asymptotically hyperbolic manifolds as critical points of the volume-renormalised mass, recently introduced by Dahl, Kröncke, and McCormick (A volume-renormalized mass for asymptotically hyperbolic manifolds. arXiv preprint arXiv:2307.06196, 2023). In particular, we show that critical points of the volume-renormalised mass over the space of constant scalar curvature asymptotically hyperbolic manifolds without boundary, or satisfying appropriate boundary conditions, are exactly V-static metrics. This is directly analogous to the relationship between critical points of the ADM mass and static metrics for asymptotically flat manifolds.