We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein--Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates---and hence new unique continuation results---for Klein--Gordon equations on a larger class of spacetimes, in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful---both presently and more generally beyond this article---for treating tensorial objects with asymptotic limits at the conformal boundary.

中文翻译：

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渐近反德西特时空波的零测地线和改进的独特连续性

我们考虑克莱因-戈登方程在渐近反德西特时空上的解是否可以从共形边界唯一地延续的问题。肯定的答案首先由第二作者和 G. Holzegel 在边界几何的适当假设和在足够长的时间跨度上施加的边界数据给出。关键步骤是为保形边界附近的 Klein-Gordon 算子建立 Carleman 估计。在本文中，我们进一步改进了上述结果。首先，我们建立了新的 Carleman 估计——并因此建立了新的独特的连续结果——对于更大类时空上的 Klein-Gordon 方程，特别是具有更一般的边界几何形状。其次，我们主张最优性，在许多方面，通过将我们的假设连接到共形边界附近的零测地线轨迹；这些测地线在构建独特延续的反例中起着至关重要的作用。最后，我们开发了一种新的协变形式主义，这将是有用的——无论是目前还是更广泛地超出本文——用于处理在共形边界处具有渐近限制的张量对象。