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Complex earthquakes and deformations of the unit disk

Epstein, D. B. A. and Marden, A. and Markovic, V. (2006) Complex earthquakes and deformations of the unit disk. Journal of Differential Geometry, 73 (1). pp. 119-166. ISSN 0022-040X. https://resolver.caltech.edu/CaltechAUTHORS:20170508-101430525

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Abstract

We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the λ-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasi-conformally equivalent to U, and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.4310/jdg/1146680514DOIArticle
https://projecteuclid.org/euclid.jdg/1146680514PublisherArticle
Additional Information:© 2005 International Press. Received 05/24/2005.
Issue or Number:1
Record Number:CaltechAUTHORS:20170508-101430525
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170508-101430525
Official Citation: Epstein, D.B.A.; Marden, A.; Markovic, V. Complex earthquakes and deformations of the unit disk. J. Differential Geom. 73 (2006), no. 1, 119--166. doi:10.4310/jdg/1146680514. http://projecteuclid.org/euclid.jdg/1146680514
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77254
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 May 2017 19:53
Last Modified:03 Oct 2019 17:55

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