From angled triangulations to hyperbolic structures

TL;DR

This survey explains Casson and Rivin's method for constructing hyperbolic structures on 3-manifolds by subdividing into ideal tetrahedra and solving gluing equations, highlighting an elementary proof of Rivin's theorem.

Contribution

Provides an elementary proof of Rivin's theorem linking volume functional critical points to hyperbolic structures on 3-manifolds.

Findings

Critical points of the volume functional yield hyperbolic structures.

The method decomposes the problem into linear and non-linear parts.

The proof simplifies understanding of hyperbolic structure existence.

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.