Type
Text
Type
Dissertation
Advisor
Anderson, Michael | Khuri, Marcus | Chen, Xiu-Xiong | Rocek, Martin.
Date
2013-12-01
Keywords
Mathematics | boundary, ricci flow
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76395
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
In this thesis, we investigate issues related to boundary value problems for the Ricci flow. First, we focus on a compact manifold with boundary and show the short-time existence, regularity and uniqueness of the flow. To obtain these results we impose the boundary conditions proposed by Anderson for the Einstein equations, namely the mean curvature and the conformal class of the boundary. We also show that a certain continuation principle holds. Our methods still apply when the manifold is not compact, as long as it has compact boundary and an appropriate control of the geometry at infinity. Secondly, motivated by the static extension conjecture in Mathematical General Relativity, we study a boundary value problem for the Ricci flow on warped products. We impose the boundary data proposed by Bartnik for the static vacuum equations, which are the mean curvature and the induced metric of the boundary of the base manifold. We conclude the thesis applying the results above to study the flow on a 3-manifold with symmetry. We show the long time existence of the flow and study its behavior in different situations. | 84 pages
Recommended Citation
Gianniotis, Panagiotis, "The Ricci flow on manifolds with boundary" (2013). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2318.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2318