<p>We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality. We also prove the ZAS inequality for graphs in Minkowski space. Furthermore, we define a new quasi-local mass functional and show that it satisfies certain desirable properties.</p> / Dissertation
| Identifer | oai:union.ndltd.org:DUKE/oai:dukespace.lib.duke.edu:10161/3857 |
| Date | January 2011 |
| Creators | Lam, Mau-Kwong George |
| Contributors | Bray, Hubert L |
| Source Sets | Duke University |
| Detected Language | English |
| Type | Dissertation |
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