We prove a local well-posedness theorem for the (n+1)-dimensional Einstein equations in Lorentzian signature, with initial data $(\tilde g, K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\hat g$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an n-dimensional asymptotically hyperbolic Riemannian manifold $(M,\tilde g)$ such that the conformally rescaled metric $x^2 \tilde g$ (with $x$ a boundary defining function) extends to the closure $\bar M$ of $M$ as a metric of class $C^{n-1}$ which is also polyhomogeneous of class $C^{p}$ on $\bar M$. Likewise we assume that the conformally rescaled symmetric (0,2)-tensor $x^{2}K$ extends to the closure as a tensor field of class $C^{n-1}$ which is polyhomogeneous of class $C^{p-1}$. We assume that the initial data $(\tilde g, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M\times (-T_0,T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension n, such that if $p \geq 2q+r_n$, with $q$ a positive integer, then there is $T>0$, depending only on the norms of the initial and boundary data, such that the Einstein equations have a unique (up to a diffeomorphism) solution $g$ on $(-T,T)\times M$ with the above initial and boundary data, which is such that $x^2g$ is of class $C^{n-1}$ and polyhomogeneous of class $C^q$. Furthermore, if $x^2\tilde g$ and $x^2K$ are polyhomogeneous of class $C^\infty$ and $\hat g$ is in $C^\infty$, then $x^2g$ is polyhomogeneous of class $C^\infty$.

中文翻译：

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具有规定共形无穷大的洛伦兹爱因斯坦度量

我们证明了 Lorentzian 签名中 (n+1) 维爱因斯坦方程的局部适定性定理，初始数据 $(\tilde g, K)$ 在无穷远处的渐近几何类似于反德西特 (AdS ) 空间，兼容边界数据 $\hat g$ 规定在时空的类时间共形边界。更准确地说，我们考虑一个 n 维渐近双曲黎曼流形 $(M,\tilde g)$ 使得共形重新缩放的度量 $x^2 \tilde g$（其中 $x$ 是一个边界定义函数）扩展到闭包$M$ 的 $\bar M$ 作为 $C^{n-1}$ 类的度量，它也是 $\bar M$ 上 $C^{p}$ 类的多齐次。同样，我们假设共形重新缩放的对称 (0,2)-张量 $x^{2}K$ 作为类 $C^{n-1}$ 的张量场扩展到闭包，它是类 $C^ 的多齐次{p-1}$。我们假设初始数据 $(\tilde g, K)$ 满足爱因斯坦约束方程，并且边界数据在 $\partial M\times (-T_0,T_0)$ 上属于 $C^p$ 类并且满足一组与初始数据的自然相容性条件。然后我们证明存在一个整数$r_n$，只依赖于维度n，使得如果$p\geq 2q+r_n$，其中$q$为正整数，则存在$T>0$，仅依赖于在初始和边界数据的范数上，使得爱因斯坦方程在具有上述初始和边界数据的 $(-T,T)\times M$ 上具有唯一的（直至微分同胚）解 $g$，即使得 $x^2g$ 属于 $C^{n-1}$ 类并且是 $C^q$ 类的多齐次。此外，如果 $x^2\tilde g$ 和 $x^2K$ 是 $C^\infty$ 类的多齐次且 $\hat g$ 在 $C^\infty$ 中，