The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighbourhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper we will analyse such set of equations on manifolds euclidean at infinity in a neighbourhood of asymptotically euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parametrize an open subset in the space of solutions of the ECE. Also, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.

中文翻译：

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欧几里得无穷大流形和闭流形上的简化薄夹层方程：存在性和多重性

简化的薄夹层方程 (RTSE) 出现在惠勒对广义相对论的爱因斯坦约束方程 (ECE) 的薄夹层方法中。众所周知，这些方程通常不能被适定，但是，在封闭流形上，已经建立了适定性的充分条件。特别是，已经表明 RTSE 在约束方程的脐带解的邻域中很好地设定，而没有保形 Killing 场。在本文中，我们将在 ECE 的渐近欧几里德 (AE) 解的邻域中分析关于无穷远流形欧几里德的此类方程组。这个方向的主要结论是，在承认 Yamabe 正度量的 AE 流形上，RTSE 的解参数化了 ECE 解空间中的一个开放子集。还，我们表明，在封闭流形的情况下，这些方程在 ECE 的脐带解周围适定，并提供了一些相关的例子。最后，将证明在封闭流形上的真空 ECE 的脐带解集合中，RTSE 通常是适定的。