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MiS Preprint
42/1999
Curvature estimates and the positive mass theorem
Hubert Bray and Felix Finster
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 $(R^3, \delta_{ij})$. 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 $(R^3, \delta_{ij})$, in the sense that there is an upper bound for the $L^2$ 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.
