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Minimal surfaces, hyperbolic 3-manif...

Sanders, Andrew Michael.

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Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces./
Author:	Sanders, Andrew Michael.
Description:	93 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.
Contained By:	Dissertation Abstracts International74-12B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3590665
ISBN:	9781303306198

Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces.
Sanders, Andrew Michael.

Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces. - 93 p.

Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.

Thesis (Ph.D.)--University of Maryland, College Park, 2013.

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 H , the space of representations R (pi1(Sigma), PSL2( C )), and the space M (X) of solutions to the self-duality equations on a rank-2 complex vector bundle over a Riemann surface X &sime; Sigma.

ISBN: 9781303306198Subjects--Topical Terms:

515831
Mathematics.

Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces.
LDR:02885nam a2200313 4500 001 1964764
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020 $a 9781303306198
035 $a (MiAaPQ)AAI3590665
035 $a AAI3590665
040 $a MiAaPQ $c MiAaPQ
100 1 $a Sanders, Andrew Michael. $3 2101274
245 1 0 $a Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces.
300 $a 93 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-12(E), Section: B.
500 $a Adviser: William Goldman.
502 $a Thesis (Ph.D.)--University of Maryland, College Park, 2013.
520 $a 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 H , the space of representations R (pi1(Sigma), PSL2( C )), and the space M (X) of solutions to the self-duality equations on a rank-2 complex vector bundle over a Riemann surface X &sime; Sigma.
520 $a Inside both H and R (pi1(Sigma), PSL2( C )) lies the space AF of almost-Fuchsian manifolds comprised of quasi-Fuchsian 3-manifolds M &sime; Sigma x 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.
520 $a Next, the space 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.
520 $a 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) → PSL2( R ), finally obtaining an explicit formula for a deformation arising from minimal surfaces in terms of Fuchsian and bending deformations. Interestingly, the hyperkahler structure on the moduli space M of solutions to the self-duality equations makes an appearance here.
590 $a School code: 0117.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
690 $a 0405
690 $a 0364
710 2 $a University of Maryland, College Park. $b Mathematics. $3 1266601
773 0 $t Dissertation Abstracts International $g 74-12B(E).
790 $a 0117
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3590665