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We prove the well-posedness of the initial boundary value problem for theEinstein equations with sole boundary condition the requirement that thetimelike boundary is totally geodesic. This provides the first well-posednessresult for this specific geometric boundary condition and the first setting forwhich geometric uniqueness in the original sense of Friedrich holds for theinitial boundary value problem. Our proof relies on the ADM system for theEinstein vacuum equations, formulated with respect to a parallelly propagatedorthonormal frame along timelike geodesics. As an independent result, we firstestablish the well-posedness in this gauge of the Cauchy problem for theEinstein equations, including the propagation of constraints. More precisely,we show that by appropriately modifying the evolution equations, using theconstraint equations, we can derive a first order symmetric hyperbolic systemfor the connection coefficients of the orthonormal frame. The propagation ofthe constraints then relies on the derivation of a hyperbolic system involvingthe connection, suitably modified Riemann and Ricci curvature tensors and thetorsion of the connection. In particular, the connection is shown to agree withthe Levi-Civita connection at the same time as the validity of the constraints.In the case of the initial boundary value problem with totally geodesicboundary, we then verify that the vanishing of the second fundamental form ofthe boundary leads to homogeneous boundary conditions for our modified ADMsystem, as well as for the hyperbolic system used in the propagation of theconstraints. An additional analytical difficulty arises from a loss of controlon the normal derivatives to the boundary of the solution. To resolve thisissue, we work with an anisotropic scale of Sobolev spaces and exploit thespecific structure of the equations.

The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary