  Recently, there have been efforts to solve Einstein's equation in the context
of a conformal compactification of space-time. Of particular importance in this
regard are the so called CMC-foliations, characterized by spatial hyperboloidal
hypersurfaces with a constant extrinsic mean curvature K. However, although of
interest for general space-times, CMC-slices are known explicitly only for the
spherically symmetric Schwarzschild metric. This work is devoted to numerically
determining axisymmetric CMC-slices within the Kerr solution. We construct such
slices outside the black hole horizon through an appropriate coordinate
transformation in which an unknown auxiliary function A is involved. The
condition K=const. throughout the slice leads to a nonlinear partial
differential equation for the function A, which is solved with a
pseudo-spectral method. The results exhibit exponential convergence, as is to
be expected in a pseudo-spectral scheme for analytic solutions. As a
by-product, we identify CMC-slices of the Schwarzschild solution which are not
spherically symmetric.
