# Curvature estimates and the Positive Mass Theorem

Journal Article (Journal Article)

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 δ ). 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 , δ ), in the sense that there is an upper bound for the L 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. 3 3 2 ij ij

### Full Text

### Duke Authors

### Cited Authors

- Bray, H; Finster, F

### Published Date

- January 1, 2002

### Published In

### Volume / Issue

- 10 / 2

### Start / End Page

- 291 - 306

### International Standard Serial Number (ISSN)

- 1019-8385

### Digital Object Identifier (DOI)

- 10.4310/CAG.2002.v10.n2.a3

### Citation Source

- Scopus