  We present a set of boundary conditions for solving the elliptic equations in
the Initial Data Problem for space-times containing a black hole, together with
a number of constraints to be satisfied by the otherwise freely specifiable
standard parameters of the Conformal Thin Sandwich formulation. These
conditions altogether are sufficient for the construction of a horizon that is
instantaneously in equilibrium in the sense of the Isolated Horizons formalism.
We then investigate the application of these conditions to the Initial Data
Problem of binary black holes and discuss the relation of our analysis with
other proposals that exist in the literature.
