to the intrinsic metric and the second fundamental form induced by the prospective space-time solution $g$ on the embedded hypersurface ${\cal S}$. They will thus have to satisfy the constraints implied by the field equations on space-like hypersurfaces. There are known now fairly general methods to construct solutions to the vacuum constraints ([@bartnik:isenberg], [@holst:nagy:2008], [@maxwell]).

The consistency conditions to be satisfied on $\Sigma$ may create difficulties though. If the intended application requires us to follow the procedure (i), the data will need to satisfy differential relations at any order on $\Sigma$ (the order will be finite if the differentiability requirements on the solutions are relaxed). Such data cannot be constructed by the conformal standard method, in which the problem of solving the constraints is immediately reduced to one of solving elliptic equations. More recent methods show more flexibility in exploiting the underdeterminedness of the constraints ([@chrusciel:delay:2003], [@corvino], [@corvino:schoen]). But even if they turn out to be useful in analytical studies of the present problem there may remain challenges for the numerical relativist.

If the intended application allows us to follow the procedure (ii), it turns out that things are easier also in the case of Einstein’s field equations. In the 