Type
Text
Type
Dissertation
Advisor
Anderson, Michael T. | Marcus A. Khuri | Daryl Geller | Peter van Nieuwenhuizen.
Date
2010-08-01
Keywords
conformally compact Einstein metrics, orbifold degeneration, Taub-bolt | Mathematics
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/11401/70999
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
In his investigation of the Dirichlet problem for conformally compact Einstein metrics, Anderson showed that there are (at most) three possibilities for the behavior, under subsequences, of a sequence of conformally compact Einstein metrics, with controlled conformal infinities, on a four-manifold: convergence, orbifold degeneration, or cusp formation. Motivated by this result, we study the phenomenon of orbifold degeneration of a curve of conformally compact Einstein metrics. We start by presenting some background material. After this, we survey the known results concerning the Dirichlet problem, and we address some open questions regarding orbifold degeneration. We then analyze a concrete example of orbifold degeneration, namely, the Taub-bolt family of conformally compact Einstein metrics on the tangent bundle of the two-sphere, and we show that the orbifold Taub-bolt metric is nondegenerate, that is, the kernel of the Bianchi gauged Einstein operator is trivial for this metric. Finally, we obtain results related to a conjecture of Anderson about the boundary of the completion, in the pointed Gromov-Hausdorff topology, of the space of conformally compact Einstein metrics on a four-manifold. These last results give necessary conditions for orbifold degeneration to occur.
Recommended Citation
Girao, Frederico Vale, "Orbifold Degeneration of Conformally Compact Einstein Metrics" (2010). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 207.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/207