Mathematics – Geometric Topology
Scientific paper
2010-04-03
Contemporary Mathematics 541 (2011), 159-182
Mathematics
Geometric Topology
24 pages, 7 figures. Survey article to appear in the Contemporary Mathematics volume entitled "Interactions Between Hyperbolic
Scientific paper
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.
Futer David
Guéritaud François
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