@article {644768, title = {Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space}, journal = {Transactions of the American Mathematical Society}, volume = {374 }, year = {2021}, pages = {3535-3555}, 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.}, url = {https://www.ams.org/journals/tran/2021-374-05/S0002-9947-2021-08297-5/}, author = {Aghil Alaee and Armando J. Cabrera Pacheco and Stephen McCormick} }