Stability Of A Quasi-Local Positive Mass Theorem For Graphical Hypersurfaces Of Euclidean Space

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 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.