258 research outputs found
Rigidity and volume preserving deformation on degenerate simplices
Given a degenerate -simplex in a -dimensional space
(Euclidean, spherical or hyperbolic space, and ), for each , , Radon's theorem induces a partition of the set of -faces into two
subsets. We prove that if the vertices of the simplex vary smoothly in
for , and the volumes of -faces in one subset are constrained only to
decrease while in the other subset only to increase, then any sufficiently
small motion must preserve the volumes of all -faces; and this property
still holds in for if an invariant of
the degenerate simplex has the desired sign. This answers a question posed by
the author, and the proof relies on an invariant we discovered
for any -stress on a cell complex in . We introduce a
characteristic polynomial of the degenerate simplex by defining
, and prove that the roots
of are real for the Euclidean case. Some evidence suggests the same
conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr
Variational properties of the discrete Hilbert-Einstein functional
This is a survey on rigidity and geometrization results obtained with the
help of the discrete Hilbert-Einstein functional, written for the proceedings
of the "Discrete Curvature" colloquium in Luminy.Comment: 10 page
Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds
We show that the hyperbolic structure on a closed, orientable, hyperbolic
3-manifold can be constructed from a solution to the hyperbolic gluing
equations using any triangulation with essential edges. The key ingredients in
the proof are Thurston's spinning construction and a volume rigidity result
attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we
show that this gives a new algorithm to detect hyperbolic structures on closed
3-manifolds.Comment: 17 page
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