低正则性广义相对论的类空特征Cauchy问题
The spacelike-characteristic Cauchy problem of general relativity in low regularity
摘要：
本文研究了Einstein真空方程的类空特征Cauchy问题。我们证明了给定的初始数据上的一个最大的紧凑类空超曲面$Σ\simeq \overline{B(0,1)} \subset \mathbb{R}^3$和传出的零超曲面$\mathcal{H}$发出的$\partial Σ$，爱因斯坦真空方程的解的存在时间控制的低正则性边界上的初始数据的曲率水平在$L^2$。证明中使用了Klainerman，Szeftel和Rodnianski的有界$L^2$曲率定理，Czimek的约束方程的延拓方法，Czimek发展的低正则性Cheeger-Gromov理论，Czimek和Graf的低正则性零超曲面上的正则叶理，以及类空极大超曲面的整体椭圆估计.
英文摘要：
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $Σ\simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and the outgoing null hypersurface $\mathcal{H}$ emanating from $\partial Σ$, the time of existence of a solution to the Einstein vacuum equations is controlled by low regularity bounds on the initial data at the level of curvature in $L^2$. The proof uses the bounded $L^2$ curvature theorem by Klainerman, Szeftel and Rodnianski, the extension procedure for the constraint equations by Czimek, Cheeger-Gromov theory in low regularity developed by Czimek, the canonical foliation on null hypersurfaces in low regularity by Czimek and Graf, and global elliptic estimates for spacelike maximal hypersurfaces.
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