Curvature Estimates and the Positive Mass Theorem
Abstract
The Positive Mass Theorem implies 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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 1999
- DOI:
- 10.48550/arXiv.math/9906047
- arXiv:
- arXiv:math/9906047
- Bibcode:
- 1999math......6047B
- Keywords:
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- Differential Geometry;
- Mathematical Physics
- E-Print:
- 12 pages, LaTeX (published version)