Stability of Euclidean 3-space for the positive mass theorem

Abstract

We show that the Euclidean 3-space R3 is stable for the Positive Mass Theorem in the following sense. Let (Mi,gi) be a sequence of complete asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that the ADM mass m(gi) of one end of Mi converges to 0. Then for all i, there is a subset Zi in Mi such that Mi∖Zi contains the given end, the area of the boundary ∂Zi converges to zero, and (Mi∖Zi,gi) converges to R3 in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of ∂Zi in terms of m(gi). As an application of the main result, we also prove R. Bartnik’s strict positivity conjecture.

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