When AdS$_3$ Grows Hair: Boson Stars, Black Holes, and Double-Trace Deformations

We analyse three-dimensional Einstein gravity coupled to a massive complex scalar field with double-trace boundary conditions. Using high-precision spectral methods, we construct regular AdS$_3$ boson stars together with axisymmetric and non-axisymmetric hairy black holes. For each azimuthal number $m$, the hairy black holes bifurcate from the BTZ family at the corresponding double-trace instability onset. When the double-trace parameter satisfies $\kappa < \kappa_{\rm AdS}$, global AdS$_3$ becomes unstable and we identify its nonlinear endpoint as a zero-frequency boson star with energy below that of AdS$_3$, thereby providing the true ground state of the theory. In the microcanonical ensemble, hairy black holes always carry greater entropy than BTZ at fixed mass and angular momentum, and thus dominate whenever they exist. With notable exceptions, typically hairy black holes do not dominate the canonical nor the grand-canonical ensembles. We further show that, in the singular extremal limit, axisymmetric black holes saturate a generalised minimum-energy theorem under double-trace boundary conditions. These results yield the full nonlinear phase diagram of AdS$_3$ gravity with double-trace deformations.