Unique continuation through compact hypersurfaces

广义相对论中的一个重要数学问题是静止黑洞的唯一性猜想。现有的逼近猜想的方法都是基于洛伦兹流形上双曲型偏微分方程解的唯一延拓结果。现有的大多数唯一延拓结果的主要限制是它们是局部公式化的，并且没有考虑全局几何。因此，我们考虑如下问题：给定一个线性齐次偏微分方程解在紧超曲面的一侧消失，它必然在超曲面的开邻域上消失吗？鉴于我们以前的工作，对这个问题的深入理解将是朝着黑洞唯一性猜想迈出的重要一步。我们将从微局域分析的角度来处理这个问题。

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.