Initial data for perturbed Kerr black holes on hyperboloidal slices

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.