embedded space-like hypersurface (with boundary $\Sigma$) in a space-time which is given ([@bartnik:1984], [@bartnik:1988]). In an initial value problem the embedding of the initial hypersurface is constructed, however, simultaneously with the space-time and the roles of the different parts of the data and the constraints cannot be seperated from each other so clearly. Nevertheless, we shall discuss the boundary data in a similar way to illustrate what has to be achieved.

The boundary conditions and data must control the evolution of the boundary. In [@friedrich:nagy] this is achieved by prescribing on the boundary ${\cal T}$ its prospective mean extrinsic curvature and on the edge $\Sigma$ a direction tangential to ${\cal T}$ and transverse to $\Sigma$. The coordinate $x^3$ is then defined so that ${\cal T} = \{x^3 = 0\}$. In [@kreiss:et:al:2009] this has not been completely analysed yet. We note that an understanding of the mechanism which controls the boundary becomes important if the boundary developes a tendency to form cusps or selfintersections.

The boundary conditions have to be given such that constraints and/or gauge conditions are preserved. This requirement, which is taken care of in quite different ways in [@friedrich:nagy] and [@kreiss:et:al:2009] poses considerable difficulties. In an approach based 