Citation:
Alaee, A., Pacheco, A.J.C. & McCormick, S. Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space. Transactions of the American Mathematical Society 374 , 3535-3555 (2021).
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.