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Geometric transitions : from hyperbolic to Ads geometry

Author: Jeffrey Edward Danciger; Steve Kerckhoff; G Carlsson; Maryam Mirzakhani; Stanford University. Department of Mathematics.
Publisher: 2011.
Dissertation: Ph. D. Stanford University 2011
Edition/Format:   Thesis/dissertation : Document : Thesis/dissertation : eBook   Computer File : English
Database:WorldCat
Summary:
We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds  Read more...
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Details

Material Type: Document, Thesis/dissertation, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Jeffrey Edward Danciger; Steve Kerckhoff; G Carlsson; Maryam Mirzakhani; Stanford University. Department of Mathematics.
OCLC Number: 743406573
Notes: Submitted to the Department of Mathematics.
Description: 1 online resource
Responsibility: Jeffrey Danciger.

Abstract:

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We also discuss combinatorial/algebraic tools for constructing transitions using ideal tetrahedra. Using these tools we prove that transitions can always be constructed when the underlying manifold is a punctured torus bundle.

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