Ideal Hyperbolic Polyhedra and Discrete Uniformization
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Ideal Hyperbolic Polyhedra and Discrete Uniformization Boris Springborn1 Received: 28 March 2018 / Revised: 7 August 2019 / Accepted: 22 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces Tg,n of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over Tg,n , and invariant under the action of the mapping class group. Keywords Decorated Teichmüller space · Penner coordinates · Horocycle · Discrete conformal equivalence · Triangulated surface Mathematics Subject Classification 57M50 · 52B10 · 52C26
1 Introduction This article is concerned with two types of problems that are in fact equivalent: realization problems for ideal hyperbolic polyhedra with prescribed intrinsic metric, and discrete uniformization problems. We develop a variational method to prove the respective existence and uniqueness theorems. Special attention is paid to the case of genus zero, because it turns out to be the most difficult one. In particular, we provide a constructive variational proof of Rivin’s realization theorem for convex ideal polyhedra with prescribed intrinsic metric:
Editor in Charge: Kenneth Clarkson Boris Springborn [email protected] 1
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
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Discrete & Computational Geometry
Theorem 1.1 (Rivin [31]) Every complete hyperbolic surface S of finite area that is homeomorphic to a punctured sphere can be realized as a convex ideal polyhedron in three-dimensional hyperbolic space H 3 . The realization is unique up to isometries of H 3 . The realizing polyhedron is allowed to degenerate to a two-sided ideal polygon. The uniqueness statement of Theorem 1.1 implies that this is the case if and only if S admits an orientation reversing isometry mapping each cusp to itself. An analogous realization result for convex Euclidean polyhedra was proved by Alexandrov [2, pp. 99–100], and Rivin’s original proof of Theorem 1.1 follows the general approach introduced by Alexandrov: First, show that the realization is unique if it exists. Then use this rigidity result to show that the space of realizable metrics is open and closed in the connected space of all metrics. This topological argument does not provide a method of actually constructing a polyhedron with prescribed intrinsic metric, and to find such a method was posed as a problem for further research [31]. The proof of Theorem 1.1 presented here is variational in nature. It proceeds by transforming the realization problem into
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