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Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined onsome open subset $A \subseteq \mathbb{R}^m$, it is a classical problem ofanalysis to investigate whether the function can be extended to all of$\mathbb{R}^m$ in a certain regularity class. If an extension exists and iscontinuous, then certainly it is uniquely determined on the closure of $A$. Asimilar problem arises in general relativity for Lorentzian manifolds insteadof functions on $\mathbb{R}^m$. It is well-known, however, that even if theextension of a Lorentzian manifold $(M,g)$ is analytic, various choices are ingeneral possible at the boundary. This paper establishes a uniqueness conditionfor extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focuson low regularities: any two extensions which are anchored by an inextendiblecausal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma$ has limitpoints in both extensions, must agree locally around those limit points on theboundary as long as the extensions are at least locally Lipschitz continuous.We also show that this is sharp: anchored extensions which are only H\"oldercontinuous do in general not enjoy this local uniqueness result.

Uniqueness and non-uniqueness results for spacetime extensions