along the boundary ${\cal T}$ is tightly related to the evolution process.

ii\) Together with the reduced equations the Cauchy data on ${\cal S}$ give us control on the geometry at all orders on ${\cal S}$. In contrast, only very little direct information on the geometry of the boundary ${\cal T}$ is provided by the boundary data and conditions. In general, neither the induced metric nor the second fundamental form is available there before the solution has been determined.

The difficulties of controlling the preservation of constraints and gauge conditions which result from these properties have been overcome. There remains, however, the problem of geometric uniqueness.

An approach based on the Bianchi equation
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To illustrate the nature of the difficulties with geometric uniqueness we need to consider the initial boundary value problem for Einstein’s vacuum equation in some detail. We concentrate on those features which are specific to the treatment of the boundary.

Formulation of the PDE problem
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In the approach of [@friedrich:nagy] the Einstein equations are expressed in terms of the following unknows

$-$ the coefficients $e^{\mu}\,_k$ of a frame $\{e_k\}_{k= 0, 1, 2, 3}$ in suitable coordinates $x^{\mu}$, $\mu = 0, 1, 2, 3$,\
where the 