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

On a spacetime positive mass theorem with corners

“…where ω ± := π ± (•, ν). Furthermore if M ext is diffeomorphic to the complement of a compact set in R 3 , then M is diffeomorphic to R 3 and the data set arises as the graph of a linear combination of spacetime harmonic functions in Minkowski space.…”

“…If W(Σ) = 0, then Σ is connected, H 2 (Ω, Z) = 0 and N is a flat spacetime over Ω. In particular, if furthermore Ω ext is diffeomorphic to the complement of a compact set in a domain of R 3 , then Ω is diffeomorphic to a domain in R 3 and can be isometrically embedded into Minkowski space.…”