Unique continuation through compact hypersurfaces

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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.