论文标题
爱因斯坦方程的散装对应关系在渐近的反de安静时间
The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
论文作者
论文摘要
在本文中，我们认为带有共形边界$（\ Mathscr {i}，\ Mathfrak {g}）$的真空渐近抗DE抗De Sitter spacetimes $（\ Mathscr {m}，g）$。我们在此类空位及其在$ \ Mathscr {i} $上的共形边界数据之间建立了一个附近的信件，即$ \ mathscr {i} $。更具体地说，给定一个域$ \ mathscr {d} \ subset \ mathscr {i} $，我们证明了系数$ \ mathfrak {g}^{（0）} = \ mathfrak {g} Fefferman-Graham从边界的公制$ g $扩展，唯一确定$ g $接近$ \ mathscr {d} $，提供了$ \ mathscr {d} $满足广义的null covexity条件（gncc）。 The GNCC is a conformally invariant criterion on $\mathscr{D}$, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in $\mathscr{M}$ near $\mathscr{D}$, and with the pseudoconvexity degenerating in the limit at $ \ mathscr {d} $。作为此结果的必然性，我们推断出$（\ Mathfrak {g}^{（0）} $（\ Mathfrak {g}^{（n）}）$的同形对称性$ \ mathscr {d} $。不需要任何分析性假设的证明依赖于三种关键成分：（1）为此设置开发的垂直张量场的演算； （2）一种新型的运输和波动方程系统，用于度量和曲率量的差异； （3）最近确定了Carleman估计在保形边界附近的紧张波方程。
In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes $( \mathscr{M}, g )$ with conformal boundary $( \mathscr{I}, \mathfrak{g} )$. We establish a correspondence, near $\mathscr{I}$, between such spacetimes and their conformal boundary data on $\mathscr{I}$. More specifically, given a domain $\mathscr{D} \subset \mathscr{I}$, we prove that the coefficients $\mathfrak{g}^{(0)} = \mathfrak{g}$ and $\mathfrak{g}^{(n)}$ (the undetermined term or stress energy tensor) in a Fefferman-Graham expansion of the metric $g$ from the boundary uniquely determine $g$ near $\mathscr{D}$, provided $\mathscr{D}$ satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on $\mathscr{D}$, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in $\mathscr{M}$ near $\mathscr{D}$, and with the pseudoconvexity degenerating in the limit at $\mathscr{D}$. As a corollary of this result, we deduce that conformal symmetries of $( \mathfrak{g}^{(0)}, \mathfrak{g}^{(n)} )$ on domains $\mathscr{D} \subset \mathscr{I}$ satisfying the GNCC extend to spacetimes symmetries near $\mathscr{D}$. The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary.
