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arXiv 2201.06010gr-qcmath.AP
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渐近Anti-de Sitter时空中波的规范不变唯一延拓准则

A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes

Athanasios Chatzikaleas, Arick Shao

摘要:

我们重新考虑了一类广义张量Klein-Gordon方程的唯一延拓性质,该方程的形式为\begin{align*} \Box_{g} ϕ+ σϕ= \mathcal{G}(ϕ,\nabla ϕ) \text{,} \qquad σ\in \mathbb{R} \end{align*},在一大类渐近反德西特时空上。特别地,我们的目标是推广Holzegel,McGill和第二作者[14,15,24]的先前结果(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)我们将所谓的零凸性准则(在[24]中建立唯一连续性质所需的共形边界上的关键几何假设)通过更一般的标准也是规范不变的。(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)类似于[24],我们将广义零凸性准则的失效与共形边界附近某些零测地线的存在性联系起来。这些测地线可以用来构造唯一延拓的反例。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的唯一延拓结果的关键因素,这将在Holzegel和第二作者的即将发表的论文中讨论[16]。

英文摘要:

We reconsider the unique continuation property for a general class of tensorial Klein-Gordon equations of the form \begin{align*} \Box_{g} ϕ+ σϕ= \mathcal{G}(ϕ,\nabla ϕ) \text{,} \qquad σ\in \mathbb{R} \end{align*} on 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 [14,15,24] (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 [24] 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 [24], we connect the failure of our generalized null convexity criterion to the existence of certain null geodesics near the conformal boundary. These geodesics can be used to construct counterexamples to unique continuation. 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 [16].

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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