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arXiv · 1911.12343

Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space

Aghil Alaee·Armando J. Cabrera Pacheco·Stephen McCormick
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Abstract

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.

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Aghil Alaee, Armando J. Cabrera Pacheco, Stephen McCormick. 2020-01-21. Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space. https://arxiv.org/abs/1911.12343

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