Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index

Kerr black holes with synchronised scalar hair and azimuthal harmonic index $m>1$ are constructed and studied. The corresponding domain of existence has a broader frequency range than the fundamental $m=1$ family; moreover, larger ADM masses, $M$ and angular momenta $J$ are allowed. Amongst other salient features, non-uniqueness of solutions for fixed global quantities is observed: solutions with the same $M$ and $J$ co-exist, for consecutive values of $m$, and the ones with larger $m$ are always entropically favoured. Our analysis demonstrates, moreover, the qualitative universality of various features observed for $m=1$ solutions, such as the shape of the domain of existence, the typology of ergo-regions, and the horizon geometry, which is studied through its isometric embedding in Euclidean 3-space.