Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

Abstract: The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in $\mathbb{R}^{n+1}$. Specifically, for an asymptotically flat graphical hypersurface $M^n\subset \mathbb{R}^{n+1}$ of nonnegative scalar curvature (satisfying certain technical conditions), there is a horizontal hyperplane $\Pi\subset \mathbb{R}^{n+1}$ such that the flat distance between $M$ and $\Pi$ in any ball of radius $\rho$ can be bounded purely in terms of $n$, $\rho$, and the mass of $M$. In particular, this means that if the masses of a sequence of such graphs approach zero, then the sequence weakly converges (in the sense of currents, after a suitable vertical normalization) to a flat plane in $\mathbb{R}^{n+1}$. This result generalizes some of the earlier findings of the second author and C. Sormani and provides some evidence for a conjecture stated there.