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15 May 2009 On the near-equality case of the positive mass theorem
Dan A. Lee
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Duke Math. J. 148(1): 63-80 (15 May 2009). DOI: 10.1215/00127094-2009-021


The positive mass conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the positive mass conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The positive mass conjecture was proved by R. Schoen and S.-T. Yau for all manifolds of dimension less than 8 (see [SY]), and it was proved by E. Witten for all spin manifolds [Wi]. In this article, we consider complete asymptotically flat manifolds of nonnegative scalar curvature which are also harmonically flat in an end. We show that, whenever the positive mass theorem holds, any appropriately normalized sequence of such manifolds whose masses converge to zero must have metrics that uniformly converge to the Euclidean metric outside a compact region. This result is an ingredient in a proof, coauthored with H. Bray, of the Riemannian Penrose inequality in dimensions less than 8 (see [BL])


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Dan A. Lee. "On the near-equality case of the positive mass theorem." Duke Math. J. 148 (1) 63 - 80, 15 May 2009.


Published: 15 May 2009
First available in Project Euclid: 22 April 2009

zbMATH: 1168.53018
MathSciNet: MR2515100
Digital Object Identifier: 10.1215/00127094-2009-021

Primary: 53C20
Secondary: 83C99

Rights: Copyright © 2009 Duke University Press


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Vol.148 • No. 1 • 15 May 2009
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