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

TL;DR

This paper investigates the stability of the positive mass theorem for certain graphical hypersurfaces in Euclidean space, showing that small boundary Brown--York mass implies the manifold is close to a Euclidean hyperplane.

Contribution

It introduces a quasi-local stability result for the positive mass theorem using Brown--York mass on graphical hypersurfaces.

Findings

Small Brown--York mass on boundary implies manifold is close to Euclidean hyperplane

Establishes stability in the Federer--Fleming flat distance

Focuses on graphical manifolds in Euclidean space

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.