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

Alaee, Aghil; Cabrera Pacheco, Armando; McCormick, Stephen

doi:10.1090/tran/8297

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

Authors: Aghil Alaee, Armando J. Cabrera Pacheco and Stephen McCormick
Journal: Trans. Amer. Math. Soc. 374 (2021), 3535-3555
MSC (2020): Primary 53C20; Secondary 83C99
DOI: https://doi.org/10.1090/tran/8297
Published electronically: February 23, 2021
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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 -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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