Some stability results of positive mass theorem for uniformly asymptotically flat 3-manifolds

In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat 3-manifolds (Mi,gi) with nonnegative scalar curvature and ADM mass m(gi) tending to zero, by subtracting some open subsets Zi, whose boundary area satisfies Area(∂Zi)≤Cm(gi)^12-ε, for any base point pi∈Mi\Zi, (Mi\Zi,gi,pi) converges to the Euclidean space (R³,gE,0) in the C⁰ modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then (Mi,gi,pi) converges to (R³,gE,0) in the pointed Gromov–Hausdorff topology.

Duke Scholars

Author Conghan Dong Mathematics