[Paper Review] Volume and rigidity of hyperbolic polyhedral $3$-manifolds

Feng Luo, Tian Yang|arXiv (Cornell University)|Apr 22, 2014

Geometric and Algebraic Topology22 references3 citations

TL;DR

This paper establishes rigidity results for hyperbolic polyhedral 3-manifolds built from ideal and hyper-ideal tetrahedra by showing that their geometry is uniquely determined by curvature. Using a Fenchel dual of the volume function and convex extensions of co-volume functions, it proves that decorated and hyper-ideal hyperbolic metrics are rigid up to isometry and decoration changes, and characterizes maximum volume angle structures via edge length assignments.

ABSTRACT

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to isometry by its curvature and a decorated ideal hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. The main tool used in the proof is the Fenchel dual of the volume function.

Motivation & Objective

• To establish infinitesimal and global rigidity results for hyperbolic cone metrics on 3-manifolds constructed from ideal and hyper-ideal tetrahedra.
• To characterize maximum volume angle structures in the space of non-negative angle structures, including cases where some angles are zero.
• To extend the volume optimization program of Casson and Rivin to hyper-ideal and decorated ideal settings using convex duality.
• To prove that curvature uniquely determines the isometry class of hyper-ideal and decorated hyperbolic polyhedral metrics.
• To show that maximum volume angle structures correspond exactly to geometric realizations via edge length assignments, even when some angles vanish.

Proposed method

• Utilizes the Fenchel dual of the volume function to transform the volume optimization problem into a convex dual setting.
• Constructs a C¹-smooth convex extension of the co-volume function (dual to volume) defined on edge lengths, overcoming non-convexity of the original domain.
• Applies Legendre transformation and variational principles based on the Schl"afli formula to relate changes in volume to changes in dihedral angles and edge lengths.
• Employs Schlaefli's formula to derive differential conditions on volume variation, proving concavity of volume along line segments in the angle space.
• Uses convex analysis and lower semicontinuity to equate the Fenchel dual of co-volume with the maximum volume functional over angle structures.
• Introduces and analyzes hyper-ideal and decorated ideal angle structures, defining conditions under which edge length assignments realize maximum volume configurations.

Experimental results

Research questions

• RQ1 Can a hyper-ideal hyperbolic polyhedral metric be uniquely reconstructed from its curvature distribution?
• RQ2 To what extent do curvature and edge length data determine the isometry class of decorated ideal hyperbolic polyhedral metrics?
• RQ3 Under what conditions does a non-negative angle structure on a triangulated 3-manifold achieve maximum volume?
• RQ4 How do edge length assignments relate to angle structures that maximize volume, especially when some angles are zero?
• RQ5 Can the volume optimization program of Casson and Rivin be extended to hyper-ideal and decorated ideal tetrahedra with degenerate configurations?

Key findings

• A decorated hyperbolic polyhedral metric on a triangulated compact pseudo 3-manifold is uniquely determined up to isometry and change of decoration by its curvature distribution.
• A hyper-ideal hyperbolic polyhedral metric is uniquely determined up to isometry by its curvature, without ambiguity from decoration changes.
• Maximum volume non-negative angle structures on a triangulated closed pseudo 3-manifold correspond exactly to geometric realizations via edge lengths: positive angles arise from decorated ideal tetrahedra, and zero-angle configurations satisfy a specific inequality involving edge length exponentials.
• For hyper-ideal type angle structures, maximum volume is achieved precisely when the edge length assignment satisfies geometric realizability conditions—specifically, when zero-angle tetrahedra do not correspond to any valid hyper-ideal tetrahedron.
• The Fenchel dual of the co-volume function coincides with the maximum volume functional over the space of angle structures, establishing a duality between edge lengths and curvature.
• The critical point of the co-volume function corresponds to a geometric metric realizing the maximum volume, and such a metric exists for any curvature vector in the relative interior of the curvature domain.