A Gauge-Invariant Unique Continuation Criterion for Waves in Asymptotically Anti-de Sitter Spacetimes
A Gauge-Invariant Unique Continuation Criterion for Waves in Asymptotically Anti-de Sitter Spacetimes
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DOI：
10.1007/s00220-022-04434-6
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发表时间：
2022-07-06
影响因子：
2.4
通讯作者：
Shao, Arick
中科院分区：
文献类型：
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作者：
Chatzikaleas, Athanasios;Shao, Arick
我们重新考虑了一般类张量Klein-Gordon方程的唯一延拓性质，方程的形式为square（g）phi + sigma phi = G（phi，del phi），sigma是罗恩的一个元素，而Ron是一个大类渐近反de Sitter时空.特别地，我们的目标是推广Holzegel，麦吉尔和第二作者的先前结果（Holzegel and Shao in Commun Math Phys 347（3）：723-775，2016; Commun Partial Differ Equ 42（12）：1871-1922，2017;麦吉尔and Shao in Class Quantum Gravity 38（5）：054001，2021）（其通过在共形边界附近的新颖Carleman估计建立了上述唯一的连续性质）以如下方式：（1）我们用一个更一般的标准来代替所谓的零凸性标准-麦吉尔和Shao（2021）中建立唯一连续性质所需的共形边界上的关键几何假设-也是规范不变的。(2)我们的新的独特的延续属性可以适用于从一个更大的，更一般的类域的共形边界。(3)与麦吉尔和Shao（2021）类似，我们将广义准则的失败与共形边界附近某些零测地线的存在性联系起来。这些测地线与唯一延拓的经典Alinhac-Baouendi反例密切相关（Alinhac and Baouendi in Math Z 220（4）：561-568，1995）。最后，我们的规范不变准则和Carleman估计将构成证明完全非线性爱因斯坦真空方程唯一延拓结果的关键因素，这将在Holzegel和第二作者即将发表的论文中讨论（Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes（in preparation），2022）。
We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the formsquare(g)phi + sigma phi = G(phi, del phi), sigma is an element of Ron a large class of asymptotically anti-de-Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author (Holzegel and Shao in Commun Math Phys 347(3):723-775, 2016; Commun Partial Differ Equ 42(12):1871-1922, 2017; McGill and Shao in Class Quantum Gravity 38(5):054001, 2021) (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways:(1) We replace the so-called null convexity criterion-the key geometric assumption on the conformal boundary needed in McGill and Shao (2021) to establish the unique continuation properties-by a more general criterion that is also gauge invariant.(2) Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary.(3) Similar to McGill and Shao (2021), we connect the failure of our generalized criterion to the existence of certain null geodesics near the conformal boundary. These geodesics are closely related to the classical Alinhac-Baouendi counterexamples to unique continuation (Alinhac and Baouendi in Math Z 220(4):561-568, 1995).Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author (Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes (in preparation), 2022).