Unique continuation through compact hypersurfaces

广义相对论中一个重要的数学问题是静止黑洞的唯一性猜想。目前逼近该猜想的方法是基于洛伦兹流形上双曲型偏微分方程的唯一延拓结果。现有的大多数唯一延拓结果的主要限制是它们是局部表述的，而没有考虑全局几何。因此，我们考虑以下问题：给定一个线性齐次偏微分方程的解，它在紧致超曲面的一侧消失，它是否必然在超曲面的开放邻域中消失？鉴于我们之前的工作，对这个问题的更深入理解将是迈向黑洞唯一性猜想的重要一步。我们将从微局部分析的角度来探讨这个问题。

An important mathematical problem in general relativity is the uniqueness conjecture for stationary black holes. Current methods to approach the conjecture are based on unique continuation results for hyperbolic partial differential equations on Lorentzian manifolds. The main limitation in most existing unique continuation results is that they are formulated locally and do not take the global geometry into account. We therefore consider the following problem: Given a solution to a linear homogeneous partial differential equation which vanishes on one side of a compact hypersurface, does it necessarily vanish on an open neighbourhood of the hypersurface? A deeper understanding of this problem would be an important step towards the black hole uniqueness conjecture, in view of our previous work. We will approach this problem from the viewpoint of microlocal analysis.