LORENTZIAN EINSTEIN METRICS WITH PRESCRIBED CONFORMAL INFINITY
LORENTZIAN EINSTEIN METRICS WITH PRESCRIBED CONFORMAL INFINITY
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DOI：
10.4310/jdg/1563242472
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发表时间：
2019-07-01
影响因子：
2.5
通讯作者：
Kamran, Niky
中科院分区：
文献类型：
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作者：
Enciso, Alberto;Kamran, Niky
本文证明了（n + 1）维洛伦兹签名爱因斯坦方程的局部适定性定理，其中初始数据（g）在波浪线上，K在无穷远处的渐近几何与反德西特（AdS）空间相似，相容边界数据（g）在时空的类时共形边界上.更精确地说，我们考虑一个n维渐近双曲黎曼流形（M，（g）over tilde），使得共形重标度量x（2）（g）在tilde上（其中x是边界定义函数）扩展到M的bar上的闭包（M），作为Cn-1类度量（（M）over bar），它也是C-Polyhom（p）类（（M）over bar）的多齐次度量。同样地，我们假设共形重标对称（0，2）-张量x（2）K作为Cn-1（（M）over bar）类的张量场延拓到（M）上，它是C-Polyhom（p-1）（（M）over bar）类的多齐次张量场.我们假定初始数据（g）满足Einstein约束方程，边界数据是C-P类的偏导数Mx（-T-0，T-0），并与初始数据满足一组自然相容条件。然后我们证明了存在一个整数r（n），只依赖于维数n，使得如果p >= 2 q + r（n），q是正整数，则存在T > 0，只依赖于初始和边界数据的范数，使得爱因斯坦方程（1.1）具有唯一性（直到一个同构）具有上述初始和边界数据的关于（-T，T）xM的解g，其使得x（2）g是Cn-1（（-T，T）x（M）over bar）boolean AND C-polyhom（q）（（-T，T）x（M）over bar）.此外，若x（2）（g）在波浪线上，x（2）K是C类无穷多齐次函数，且（g）在帽上在C类无穷大（（-T-0，T-0）x偏导数M）中，则x（2）g在C类无穷多齐次函数（（-T，T）x（M）在棒上）中.
We prove a local well-posedness theorem for the (n + 1)-dimensional Einstein equations in Lorentzian signature, with initial data ((g) over tilde, K) whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data (g) over cap prescribed at the time-like conformal boundary of space-time. More precisely, we consider an n-dimensional asymptotically hyperbolic Riemannian manifold (M, (g) over tilde) such that the conformally rescaled metric x(2)(g) over tilde (with x a boundary defining function) extends to the closure (M) over bar of M as a metric of class Cn-1 ((M) over bar) which is also polyhomogeneous of class C-Polyhom(p)((M) over bar). Likewise we assume that the conformally rescaled symmetric (0, 2)-tensor x(2) K extends to (M) over bar as a tensor field of class Cn-1((M) over bar) which is polyhomogeneous of class C-Polyhom(p-1)((M) over bar). We assume that the initial data ((g) over tilde, K) satisfy the Einstein constraint equations and also that the boundary datum is of class C-P on partial derivative M x (-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 >= 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 (1.1) has a unique (up to a diffeomorphism) solution g on (-T, T) x M with the above initial and boundary data, which is such that x(2)g is an element of Cn-1((-T,T) x (M) over bar) boolean AND C-polyhom(q) ((-T,T) x (M) over bar). Furthermore, if x(2)(g) over tilde, x(2) K are polyhomogeneous of class C infinity and (g) over cap is in C infinity((-T-0,T-0) x partial derivative M), then x(2)g is in C-polyhom(infinity)((-T, T) x (M) over bar).