Minimal Surfaces, Hyperbolic 3-manifolds, and Related Deformation Spaces.
| dc.contributor.advisor | Goldman, William | en_US |
| dc.contributor.author | Sanders, Andrew Michael | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2013-06-28T06:18:04Z | |
| dc.date.available | 2013-06-28T06:18:04Z | |
| dc.date.issued | 2013 | en_US |
| dc.description.abstract | Given a closed, oriented, smooth surface $Sigma$ of negative Euler characteristic, the relationships between three deformation spaces of geometric structures are compared: the space of minimal hyperbolic germs $mathcal{H},$ the space of representations $mathcal{R}(pi_1(Sigma),text{PSL}_2(mathbb{C})),$ and the space $mathcal{M}(X)$ of solutions to the self-duality equations on a rank-2 complex vector bundle over a Riemann surface $Xsimeq Sigma.$ Inside both $mathcal{H}$ and $mathcal{R}(pi_1(Sigma),text{PSL}_2(mathbb{C}))$ lies the space $mathcal{AF}$ of almost-Fuchsian manifolds comprised of quasi-Fuchsian 3-manifolds $Msimeq Sigma times mathbb{R}$ which contain an immersed closed minimal surface whose principal curvatures lie in the interval $(-1,1).$ The structure of these manifolds is explored through a study of the domain of discontinuity of the associated almost-Fuchsian holonomy group. It is proved that there are no doubly degenerate geometric limits of almost-Fuchsian manifolds. Next, the space $mathcal{H}$ is studied through an analysis of a smooth real valued function which records the topological entropy of the geodesic flow arising from a minimal hyperbolic germ. Estimates on this function are obtained which culminate in a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. As a corollary we obtain a new proof of Bowen's theorem on quasi-circles: a quasi-Fuchsian group is Fuchsian if and only if the Hausdorff dimension of its limit set is equal to $1.$ Lastly, we recall a construction of Donaldson which shows how each minimal hyperbolic germ gives rise to a solution of the self-duality equations. In this context, we compare various deformations of a Fuchsian representation $pi_1(Sigma)rightarrow text{PSL}_2(mathbb{R}),$ finally obtaining an explicit formula for a deformation arising from minimal surfaces in terms of Fuchsian and bending deformations. Interestingly, the hyperk"{a}hler structure on the moduli space $mathcal{M}$ of solutions to the self-duality equations makes an appearance here. | en_US |
| dc.identifier.uri | http://hdl.handle.net/1903/14055 | |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.subject.pquncontrolled | Differential Geometry | en_US |
| dc.subject.pquncontrolled | Hyperbolic Geometry | en_US |
| dc.title | Minimal Surfaces, Hyperbolic 3-manifolds, and Related Deformation Spaces. | en_US |
| dc.type | Dissertation | en_US |
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