From angled triangulations to hyperbolic structures

David Futer; François Guéritaud

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arXiv · 1004.0440

From angled triangulations to hyperbolic structures

David Futer·François Guéritaud

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Abstract

This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to find hyperbolic shapes for the tetrahedra. The gluing equations decompose into a linear and non-linear part. The solutions to the linear equations form a convex polytope A. The solution to the non-linear part (unique if it exists) is a critical point of a certain volume functional on this polytope. The main contribution of this paper is an elementary proof of Rivin's theorem that a critical point of the volume functional on A produces a complete hyperbolic structure on M.

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BibTeX RIS

David Futer, François Guéritaud. 2010-04-03. From angled triangulations to hyperbolic structures. https://arxiv.org/abs/1004.0440

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