# On the near-equality case of the Positive Mass Theorem - Mathematics > Differential Geometry

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Abstract: The Positive Mass Conjecture states that any complete asymptotically flatmanifold of nonnnegative scalar curvature has nonnegative mass. Moreover, theequality case of the Positive Mass Conjecture states that in the abovesituation, if the mass is zero, then the Riemannian manifold must be Euclideanspace. The Positive Mass Conjecture was proved by R. Schoen and S.-T. Yau forall manifolds of dimension less than 8, and it was proved by E. Witten for allspin manifolds. In this paper, we consider complete asymptotically flatmanifolds of nonnegative scalar curvature that are also harmonically flat in anend. We show that, whenever the Positive Mass Theorem holds, any appropriatelynormalized sequence of such manifolds whose masses converge to zero must havemetrics that are uniformly converging to Euclidean metrics outside a compactregion. This result is an ingredient in a forthcoming proof, co-authored withH. Bray, of the Riemannian Penrose inequality in dimensions less than 8.

Author: ** Dan A. Lee**

Source: https://arxiv.org/