Curvature estimates and the positive mass theorem

Hubert Bray and Felix Finster

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Submission date: 21. Jun. 1999
Pages: 14
published in: Communications in analysis and geometry, 10 (2002) 2, p. 291-306 
DOI number (of the published article): 10.4310/CAG.2002.v10.n2.a3
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Abstract:
The Positive Mass Theorem implies in the case of equality (that is, when the total mass is zero) that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R³, _ij). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to (R³, _ij), in the sense that there is an upper bound for the L² norm of the Riemannian curvature tensor over the manifold except for a set of small measure. This curvature estimate allows us to extend the case of equality of the Positive Mass Theorem to include non-smooth manifolds with generalized non-negative scalar curvature, which we define.