Curvature Estimates and the Positive Mass Theorem
Abstract
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3manifold with nonnegative 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 nonnegative 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 nonsmooth manifolds with generalized nonnegative scalar curvature, which we define.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1999
 DOI:
 10.48550/arXiv.math/9906047
 arXiv:
 arXiv:math/9906047
 Bibcode:
 1999math......6047B
 Keywords:

 Differential Geometry;
 Mathematical Physics
 EPrint:
 12 pages, LaTeX (published version)