  Motivated by novel results in the theory of wave dynamics in black-hole
spacetimes, we analyze the dynamics of a massive scalar field surrounding a
rapidly rotating Kerr black hole. In particular, we report on the existence of
stationary (infinitely long-lived) regular field configurations in the
background of maximally rotating black holes. The effective height of these
scalar "clouds" above the central black hole is determined analytically. Our
results support the possible existence of stationary scalar field dark matter
distributions surrounding rapidly rotating black holes.
