Conformal Variations of Piecewise Constant Curvature Two and Three Dimensional Manifolds

Persistent Link:
http://hdl.handle.net/10150/560854
Title:
Conformal Variations of Piecewise Constant Curvature Two and Three Dimensional Manifolds
Author:
Thomas, Joseph
Issue Date:
2015
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Piecewise constant curvature manifolds are discrete analogues of Riemannian manifolds in which edge lengths play the role of the metric tensor. These triangulated manifolds are specified by two types of data: 1. Each edge in the triangulation is assigned a real-valued length. 2. For each simplex in the triangulation, there exists an isometric embedding of that simplex into one of three background geometries (Euclidean, hyperbolic, or spherical). In particular, this isometry respects the edge length data. By making the edge lengths functions of scalars, called conformal parameters, that are assigned to the vertices of the triangulation we obtain a conformal structure - that is, a parameterization of a discrete conformal class. We discuss how our definition of conformal structure places several existing notions of a discrete conformal class in a common framework. We then describe discrete analogues of scalar curvature for 2-and 3-manifolds and study how these curvatures depend on the conformal parameters. This leads us to some local rigidity theorems - we identify circumstances in which the mapping from conformal parameters to scalar curvatures is a local diffeomorphism. In three dimensions, we focus on the case of hyperbolic background geometry. We study a discrete analogue of the Einstein-Hilbert (or total scalar curvature) functional and investigate when this functional is locally convex.
Type:
text; Electronic Dissertation
Keywords:
Mathematics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Glickenstein, David

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleConformal Variations of Piecewise Constant Curvature Two and Three Dimensional Manifoldsen_US
dc.creatorThomas, Josephen
dc.contributor.authorThomas, Josephen
dc.date.issued2015en
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractPiecewise constant curvature manifolds are discrete analogues of Riemannian manifolds in which edge lengths play the role of the metric tensor. These triangulated manifolds are specified by two types of data: 1. Each edge in the triangulation is assigned a real-valued length. 2. For each simplex in the triangulation, there exists an isometric embedding of that simplex into one of three background geometries (Euclidean, hyperbolic, or spherical). In particular, this isometry respects the edge length data. By making the edge lengths functions of scalars, called conformal parameters, that are assigned to the vertices of the triangulation we obtain a conformal structure - that is, a parameterization of a discrete conformal class. We discuss how our definition of conformal structure places several existing notions of a discrete conformal class in a common framework. We then describe discrete analogues of scalar curvature for 2-and 3-manifolds and study how these curvatures depend on the conformal parameters. This leads us to some local rigidity theorems - we identify circumstances in which the mapping from conformal parameters to scalar curvatures is a local diffeomorphism. In three dimensions, we focus on the case of hyperbolic background geometry. We study a discrete analogue of the Einstein-Hilbert (or total scalar curvature) functional and investigate when this functional is locally convex.en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectMathematicsen
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorGlickenstein, Daviden
dc.contributor.committeememberKennedy, Thomasen
dc.contributor.committeememberPickrell, Douglasen
dc.contributor.committeememberGillette, Andrewen
dc.contributor.committeememberGlickenstein, Daviden
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