The geometry of asymptotically hyperbolic manifolds
Abstract/Contents
- Abstract
- We discuss the large-scale geometry of asymptotically hyperbolic manifolds. Asymptotically hyperbolic manifolds arise naturally in general relativity. However, several fundamental questions about them remain unresolved, including the asymptotically hyperbolic Penrose inequality and the static uniqueness of the Schwarzschild-anti-de Sitter metric. The main contributions of this thesis are twofold: Firstly, we introduce a new notion of renormalized volume for asymptotically hyperbolic manifolds and prove a sharp Penrose-type inequality where mass is replaced by renormalized volume. Secondly, we use the notion of renormalized volume to study isoperimetric regions in asymptotically hyperbolic manifolds. We prove that for initial data sets that are Schwarzschild-anti-de Sitter at infinity and satisfy appropriate scalar curvature lower bounds, sufficiently large coordinate spheres are uniquely isoperimetric. This is relevant in the context of Bray's isoperimetric approach to the Penrose inequality. From a geometric viewpoint, our results show that the large-scale geometry of asymptotically hyperbolic manifolds significantly differs from the more familiar asymptotically flat setting. The renormalized volume is a very different quantity from the ``mass, '' and our results suggest that it is a stronger quantity. As a consequence of this, we uncover a link between scalar curvature and the behavior of large isoperimetric regions, which is not present in the asymptotically flat setting. Additionally, we discuss isoperimetric regions in warped products and consequences for the renormalized volume of a more general class of metrics. Finally, we study rotational symmetry of expanding Ricci solitons, a problem that is formally similar to the static uniqueness question with negative cosmological constant.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2015 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Chodosh, Otis |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Brendle, Simon, 1981- |
| Primary advisor | Eichmair, Michael |
| Thesis advisor | Brendle, Simon, 1981- |
| Thesis advisor | Eichmair, Michael |
| Thesis advisor | Simon, L. (Leon), 1945- |
| Thesis advisor | White, Brian, 1957- |
| Advisor | Simon, L. (Leon), 1945- |
| Advisor | White, Brian, 1957- |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Otis Chodosh. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2015. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2015 by Otis Avram Chodosh
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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