Well-posed geometric boundary data in General Relativity, I: Conformal-mean curvature boundary data

We study the local in time well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. For conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary, well-posedness does not hold without imposing additional angle data at the corner. When the corner angle is included as corner data, we prove well-posedness of the linearized problem in $C^{\mathrm{\infty }}$, where the linearization is taken at any smooth vacuum Einstein metric.