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

我们证明了具有唯一边界条件的爱因斯坦方程初边值问题的适定性，即时间边界是完全测地线的要求。这为这个特定的几何边界条件提供了第一个适定性结果，并且提供了初始边界值问题中弗里德里希原始意义上的几何唯一性的第一个设置。我们的证明依赖于爱因斯坦真空方程的 ADM 系统，该系统是针对沿类时测地线平行传播的正交坐标系制定的。作为独立的结果，我们首先在爱因斯坦方程的柯西问题的这个规范中建立适定性，包括约束的传播。更准确地说，我们表明，通过适当修改演化方程，使用约束方程，我们可以推导出正交框架的连接系数的一阶对称双曲系统。然后，约束的传播依赖于涉及连接、适当修改的黎曼和里奇曲率张量以及连接的扭转的双曲系统的推导。特别是，该连接在约束有效性的同时与 Levi-Civita 连接一致。在具有完全测地线边界的初始边界值问题的情况下，我们然后验证边界的第二基本形式的消失导致我们修改的 ADM 系统以及传播中使用的双曲系统的齐次边界条件的约束。另一个分析困难是由于对解边界的正态导数失去控制。为了解决这个问题，我们使用 Sobolev 空间的各向异性尺度并利用方程的特定结构。