From angled triangulations to hyperbolic structures

By David Futer, Franccois Gu'eritaud. 2010

61 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/318187b731b21d9a6bc2ca5414985d1399895839

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Discrete conformal maps and ideal hyperbolic polyhedra (2010, background): “One is due to Rivin [29] (see also the recent survey article by Futer and Guéritaud [11], which provides a wealth of material that is otherwise difficult to find), and the other is again related to it by the same sort of singular Legendre duality.”
An Introduction to Geometric Topology (2016, background): “Many of the arguments of this chapter, including all the discussion on angle structures, have been taken from a nice paper of Futer – Guéritaud [21].”