Ideal Hyperbolic Polyhedra and Discrete Uniformization

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...

Full description

Saved in:
Bibliographic Details
Published inDiscrete & computational geometry Vol. 64; no. 1; pp. 63 - 108
Main Author Springborn, Boris
Format Journal Article
LanguageEnglish
Published New York Springer US 01.07.2020
Springer Nature B.V
Subjects
Combinatorics
Computational Mathematics and Numerical Analysis
Mapping
Mathematical analysis
Mathematics
Mathematics and Statistics
Polyhedra
Theorems
Variational principles
57M50
52B10
52C26
Decorated Teichmüller space
Penner coordinates
Horocycle
Discrete conformal equivalence
Triangulated surface
Online AccessGet full text
ISSN0179-5376
1432-0444
DOI10.1007/s00454-019-00132-8

Cover

More Information
Summary: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 T ~ g , n of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over T g , n , and invariant under the action of the mapping class group.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-019-00132-8