Calculus of generalized hyperbolic tetrahedra

Guo, Ren

doi:10.1007/s10711-010-9561-0

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…Since

B_{S}

is connected, the signature of the Hessian matrix

- \frac{1}{2} {[\partial l_{α} / \partial a_{β}]}_{α, β \in S}

of the volume function

v o l

B_{S}

is independent of the choice of

a \in {scriptB}_{S}

. By a direct calculation, using the formula in Guo (, Theorem 1), the matrix

- \frac{1}{2} {[\partial l_{α} / \partial a_{β}]}_{α, β \in S}

is negative definite at

a_{α} = π / 4

for each

α \in S

. This implies that

v o l

is locally strictly concave in

B_{S}

.…”

Section: Volume Maximization Of Angle Structuresmentioning

confidence: 99%

Volume and rigidity of hyperbolic polyhedral 3‐manifolds

Luo

Yang

2018

Journal of Topology

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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.

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“…Since

B_{S}

is connected, the signature of the Hessian matrix

- \frac{1}{2} {[\partial l_{α} / \partial a_{β}]}_{α, β \in S}

of the volume function

v o l

B_{S}

is independent of the choice of

a \in {scriptB}_{S}

. By a direct calculation, using the formula in Guo (, Theorem 1), the matrix

- \frac{1}{2} {[\partial l_{α} / \partial a_{β}]}_{α, β \in S}

is negative definite at

a_{α} = π / 4

for each

α \in S

. This implies that

v o l

is locally strictly concave in

B_{S}

.…”

Section: Volume Maximization Of Angle Structuresmentioning

confidence: 99%

Volume and rigidity of hyperbolic polyhedral 3‐manifolds

Luo

Yang

2018

Journal of Topology

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show abstract

Section: Proof Of Proposition 26mentioning

confidence: 99%

“…Let us denote by θ ij , 1 ≤ i < j ≤ 4, the dihedral angles of ∆. Putting together the computation of the partial derivatives ∂θ ij /∂ hk provided by the main theorem of [Guo11] and the Lobachevsky-Schläfli formula (7) we obtain where k is a negative constant. Therefore, in order to conclude we need to prove that 12 (cos θ 12 (cos θ 13 cos θ 23 + cos θ 14 cos θ 24 ) + cos θ 13 cos θ 24 + cos θ 14 cos θ 23 ) + 34 sin θ 12 sin θ 34 > 13 sin θ 12 sin θ 13 cos θ 23 + 14 sin θ 12 sin θ 14 cos θ 24 + 24 sin θ 12 sin θ 24 cos θ 14 + 23 sin θ 12 sin θ 23 cos θ 13 .…”

Section: Proof Of Proposition 26mentioning

confidence: 99%

On volumes of truncated tetrahedra with constrained edge lengths

Frigerio

Moraschini

2018

Period Math Hung

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Truncated tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of lowdimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces.The Lobachevsky-Schläfli formula neatly describes the behaviour of the volume of truncated tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every < 0, where 0 is an explicit constant, the regular truncated tetrahedron of edge length maximizes the volume among truncated tetrahedra whose edge lengths are all not smaller than .This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary.1991 Mathematics Subject Classification. 52A55 (primary); 52A38, 52B10, 57M50 (secondary).

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“…In particular, Figure 1: Generalised hyperbolic tetrahedron Jac(T ), the Jacobian of T , which is the Jacobian matrix of the edge length with respect to the dihedral angles, is such. This matrix enjoys many symmetries [15] and can be computed out of the Gram matrix of T [7]. In the present paper, we consider Jac ⋆ (T ), the dual Jacobian of a generalised hyperbolic tetrahedron T .…”

Section: Introductionmentioning

confidence: 99%

The Dual Jacobian of a Generalised Hyperbolic Tetrahedron, and Volumes of Prisms

Kolpakov¹,

Murakami²

2016

Tokyo J. Math.

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We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the corresponding volume formulae.Also, we obtain a volume formula for a hyperbolic n-gonal prism: the proof requires the above mentioned Jacobian, employed in the analysis of the edge lengths behaviour of such a prism, needed later for the Schläfli formula.

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Calculus of generalized hyperbolic tetrahedra

Cited by 4 publications

References 15 publications

Volume and rigidity of hyperbolic polyhedral 3‐manifolds

Volume and rigidity of hyperbolic polyhedral 3‐manifolds

On volumes of truncated tetrahedra with constrained edge lengths

The Dual Jacobian of a Generalised Hyperbolic Tetrahedron, and Volumes of Prisms

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