  We construct initial data corresponding to a single perturbed Kerr black hole
in vacuum. These data are defined on specific hyperboloidal ("ACMC-") slices on
which the mean extrinsic curvature K asymptotically approaches a constant at
future null infinity scri+. More precisely, we require that K obeys the Taylor
expansion K=K0 + s^4 where K0 is a constant and s describes a compactified
spatial coordinate such that scri+ is represented by s=0. We excise the
singular interior of the black hole and assume a marginally outer trapped
surface as inner boundary of the computational domain. The momentum and
Hamiltonian constraints are solved by means of pseudo-spectral methods and we
find exponential rates of convergence of our numerical solutions. Some physical
properties of the initial data are studied with the calculation of the Bondi
Mass, together with a multipole decomposition of the horizon. We probe the
standard picture of gravitational collapse by assessing a family of
Penrose-like inequalities and discuss in particular their rigidity aspects.
Dynamical evolutions are planned in a future project.
