  We consider the initial boundary value problem for the Einstein vacuum
equations in the maximal gauge, or more generally, in a gauge where the mean
curvature of a timelike foliation is fixed near the boundary. We prove the
existence of solutions such that the normal to the boundary is tangent to the
time slices, the lapse of the induced time coordinate on the boundary is fixed
and the main geometric boundary conditions are given by the 1-parameter family
of Riemannian conformal metrics on each two-dimensional section. As in the
local existence theory of Christodoulou-Klainerman for the Einstein vacuum
equations in the maximal gauge, we use as a reduced system the wave equations
satisfied by the components of the second fundamental form of the the time
foliation. The main difficulty lies in completing the above set of boundary
conditions such that the reduced system is well-posed, but still allows for the
recovery of the Einstein equations. We solve this problem by imposing the
momentum constraint equations on the boundary, suitably modified by quantities
vanishing in the maximal gauge setting. To derive energy estimates for the
reduced system at time t, we show that all the terms in the flux integrals on
the boundary can be either directly controlled by the boundary conditions or
they lead to an integral on the two-dimensional section at time t of the
boundary. Exploiting again the maximal gauge condition on the boundary, this
contribution to the flux integrals can then be absorbed by a careful trace
inequality in the interior energy.
