  Using a constrained formalism for Einstein equations in Dirac gauge, we
propose to compute excised quasistationary initial data for black hole
spacetimes in full general relativity. Vacuum spacetime settings are
numerically constructed by using the isolated horizon formalism; we especially
tackle the conformal metric part of our equations, assuming global
stationarity. We show that a no-boundary treatment can be used on the horizon
for the equation related to the conformal metric. We relate this finding to
previous suggestions in the literature, and use our results to assess the
widely used conformally flat approximation for computing black hole initial
data.
