We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown--York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown--York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense? Here we consider a class of compact $n$-manifolds with boundary that can be realized as graphs in $\mathbb{R}^{n+1}$, and establish the following. If the Brown--York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer--Fleming flat distance.

中文翻译：

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欧几里得空间图形超曲面的拟局部正质量定理的稳定性

我们提出了正质量定理稳定性的准局部版本。我们使用 Brown-York 准局部质量，因为它具有正性和刚性特性，因此可以研究这种刚性陈述的稳定性。具体来说，我们问如果某个紧凑流形边界的布朗-约克质量接近于零，那么该流形在某种意义上是否必须接近欧几里得域？这里我们考虑一类具有边界的紧凑 $n$-流形，可以在 $\mathbb{R}^{n+1}$ 中实现为图，并建立以下内容。如果这种紧凑流形边界的布朗-约克质量很小，则该流形接近于相对于费德勒-弗莱明平面距离的欧几里得超平面。