  We prove global existence of instantaneously complete Yamabe flows on
hyperbolic space of arbitrary dimension $m\geq3$. The initial metric is assumed
to be conformally hyperbolic with conformal factor and scalar curvature bounded
from above. We do not require initial completeness or bounds on the Ricci
curvature. If the initial data are rotationally symmetric, the solution is
proven to be unique in the class of instantaneously complete, rotationally
symmetric Yamabe flows.
