Localized Black Holes in AdS$_3$: What Happens Below c/12 Stays Below c/12

We construct asymptotically AdS$_3 \times$S$^3 \times$T$^4$ black holes that are localized on the S$^3$ and co-exist with the BTZ black hole at small positive energies. These black holes dominate the microcanonical ensemble for $ E\leq \frac{c}{24}\left(5\sqrt{5}-11\right)$, suggesting they could represent the endpoint of the BTZ instability at low energies. Remarkably, they also exist at negative energies, where pure Einstein gravity predicts no states and the BTZ black hole does not exist. They appear in the spectrum immediately above $-\frac{c}{12}$ (the energy of global AdS$_3$), and their entropy is a significant fraction (up to 1/2) of the entropy of the free orbifold CFT at negative energies. Our solutions exist in an energy window outside the universal predictable range of the modular bootstrap in large-$c$ CFT$_2$ and, despite their microcanonical dominance, do not dominate in the canonical ensemble. To calculate the holographic entanglement entropy of our solutions, we propose the first recipe that can be applied to arbitrary geometries asymptotic to AdS$_3$ times an internal manifold, and depend non-trivially on its coordinates. We find that our new geometries have an entanglement entropy nearly identical to that of the BTZ black hole with the same energy, despite having different horizon structures. However, they can be distinguished by non-minimal extremal surfaces, which unveil finer details of the microstructure.