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arXiv 2207.14217gr-qchep-thmath.AP
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渐近反德西特时空中爱因斯坦方程的体-边对应

The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes

Gustav Holzegel, Arick Shao

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60 pages, 1 figure. Version accepted at ARMA
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AI总结：

本文研究渐近反德西特时空的体-边对应，证明满足广义零凸性条件的边界区域上的Fefferman-Graham展开系数可唯一确定附近时空度规，且共形对称可延拓，证明无需解析假设，依赖三个关键技术工具。

摘要：

本文研究具有共形边界$( \mathscr{I}, \mathfrak{g} )$的真空渐近反德西特时空$( \mathscr{M}, g )$，建立了这类时空在$\mathscr{I}$附近与其在$\mathscr{I}$上的共形边界数据之间的对应关系。更具体地说，给定区域$\mathscr{D} \subset \mathscr{I}$，我们证明：若$\mathscr{D}$满足广义零凸性条件（GNCC），则从边界出发的度规$g$的Fefferman-Graham展开中的系数$\mathfrak{g}^{(0)} = \mathfrak{g}$和$\mathfrak{g}^{(n)}$（待定项或应力能动张量）能唯一确定$\mathscr{D}$附近的$g$。GNCC是$\mathscr{D}$上的共形不变判据，由Chatzikaleas与第二作者首次提出，它保证了$\mathscr{M}$中$\mathscr{D}$附近存在伪凸超曲面的叶状结构，且伪凸性在趋近$\mathscr{D}$的极限下退化。作为该结果的推论，我们推导出：满足GNCC的区域$\mathscr{D} \subset \mathscr{I}$上$( \mathfrak{g}^{(0)}, \mathfrak{g}^{(n)} )$的共形对称性可延拓为$\mathscr{D}$附近的时空对称性。该证明无需任何解析性假设，依赖三个关键要素：(1) 为此场景发展的垂直张量场演算；(2) 关于度规与曲率量之差的新型输运方程和波动方程组；(3) 近期建立的共形边界附近张量波动方程的Carleman估计。

英文摘要：

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.

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