The 3-dimensional hyperbolic space

2009 ◽  
pp. 227-240
Author(s):  
Francis Bonahon
Keyword(s):  
Hyperbolic Space ◽  
3 Dimensional

Stability of minimal surfaces in a 3-dimensional hyperbolic space

1981 ◽  
Vol 36 (1) ◽  
pp. 554-557 ◽  
Author(s):  
J. L. Barbosa ◽  
M. do Carmo
Keyword(s):  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
3 Dimensional

scholarly journals Volumes of Polyhedra in Non-Euclidean Spaces of Constant Curvature

2020 ◽  
Vol 66 (4) ◽  
pp. 558-679
Author(s):  
V. A. Krasnov
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Arbitrary Dimension ◽  
Difficult Problem ◽  
Combinatorial Structure ◽  
Euclidean Spaces ◽  
Spaces Of Constant Curvature ◽  
Simple Polyhedron ◽  
3 Dimensional ◽  
Combinatorial Sense

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute).In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = -1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincare model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

Meromorphic extension of the Selberg zeta function for Kleinian groups via thermodynamic formalism

1996 ◽  
Vol 119 (1) ◽  
pp. 179-190
Author(s):  
André Rocha
Keyword(s):  
Zeta Function ◽  
Hyperbolic Space ◽  
Fundamental Domain ◽  
Thermodynamic Formalism ◽  
Kleinian Groups ◽  
Selberg Zeta Function ◽  
3 Dimensional ◽  
Expanding Map

AbstractWe prove the existence of a piecewise analytic expanding map associated to certain Kleinian groups without parabolics acting in the 3-dimensional hyperbolic space. These groups have a fundamental domain ℛ with the property that the geodesic planes containing each face are part of the tesselation. We use this map together with the methods of thermodynamic formalism to give another proof that the Selberg zeta function for such groups has a meromorphic extension to ℂ.

scholarly journals Visualization of Sphere and Horosphere Packings Related to Coxeter Tilings by Simply Truncated Orthoschemes with Parallel Faces

KoG ◽  
2021 ◽  
pp. 64-71
Author(s):  
Arnasli Yahya ◽  
Jenő Szirmai
Keyword(s):  
Hyperbolic Space ◽  
Programming Language ◽  
Reflection Group ◽  
3 Dimensional ◽  
Python Programming Language ◽  
Python Programming ◽  
Interesting Structure

In this paper we describe and visualize the densest ball and horoball packing configurations to the simply truncated 3-dimensional hyperbolic Coxeter orthoschemes with parallel faces, using the results of [24]. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter reflection group. We use the Beltrami-Cayley-Klein ball model of 3-dimensional hyperbolic space H^3, the images were made by the Python programming language.

scholarly journals Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

2003 ◽  
Vol 75 (3) ◽  
pp. 271-278
Author(s):  
Shoichi Fujimori
Keyword(s):  
Minimal Surface ◽  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
3 Dimensional

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

scholarly journals Helicoidal minimal surfaces in hyperbolic space

1989 ◽  
Vol 114 ◽  
pp. 65-75 ◽  
Author(s):  
Jaime B. Ripoll
Keyword(s):  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Helicoidal Surface ◽  
Sectional Curvatures ◽  
The One ◽  
Group Of Isometries

Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

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