标题： 时空扩展的唯一性与非唯一性结果 显示英文标题

标题： Uniqueness and non-uniqueness results for spacetime extensions

摘要： 给定一个在某个开子集$A \subseteq \mathbb{R}^m$上具有某种正则性的函数$f : A \to \mathbb{R}^n$，分析中的经典问题是研究该函数是否可以在某种正则性类中扩展到$\mathbb{R}^m$的全部范围。如果存在扩展并且是连续的，那么它在$A$的闭包上一定是唯一确定的。在广义相对论中，对于洛伦兹流形而不是$\mathbb{R}^m$上的函数，会出现类似的问题。然而众所周知，即使洛伦兹流形$(M,g)$的扩展是解析的，通常在边界处仍然可能存在各种选择。 本文建立了全局双曲洛伦兹流形 $(M,g)$ 的扩展的唯一性条件，重点在于低正则性：如果两个扩展在某种意义上由一条不可延伸的因果曲线 $\gamma : [-1,0) \to M$ 锚定，即 $\gamma$ 在两个扩展中都有极限点，则只要扩展至少是局部利普希茨连续的，它们在边界上的那些极限点附近必须一致。 我们还表明，这是精确的：仅是赫尔德连续的锚定扩展通常不具有这种局部唯一性结果。 显示英文摘要

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