Hyperbolic Plane H2 Research Articles

Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H2 by curved hyperbolic equilateral triangles whose vertex angles are 2π/d for d=7,8,9,… On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H3. We also show that a regular tessellation of H3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H4. If n>4, then there exists no regular tessellation of Hn.

Geometric phases of nonlinear elastic [formula omitted]-rotors via Cartan’s moving frames

We study the geometric phases of nonlinear elastic N-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold B, and fibers F along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan’s structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold. Without lose of generality, we show that one has the freedom to define the rotation sign of the total angular momentum A of the elastic rotors as either positive or negative, e.g., counterclockwise or clockwise, respectively, or viceversa. This endows the base manifold B with two distinct metrics both compatible with the geometric phase. In particular, the metric is pseudo-Riemannian if A<0, and the shape manifold is a 2D Robertson-Walker spacetime with positive curvature. For A>0, the shape manifold is the hyperbolic plane H2 with negative curvature. We then generalize our results to free elastic N-rotors. We show that the associated shape manifold B is reducible to the product manifold of N−1 hyperbolic planes H2 (A>0), or 2D Robertson-Walker spacetimes (A<0) depending on the convection used to define the rotation sign of the total angular momentum. We then consider elastic N-rotors subject to time-dependent self-equilibrated moments. The N-dimensional shape manifold of the extended autonomous system has a structure similar to that of the (N−1)-dimensional shape manifold of free elastic rotors. The Riemannian structure of the shape manifold provides an intrinsic measure of the closeness of one shape to another in terms of curvature, or induced geometric phase.

Horocyclic Brunn-Minkowski inequality

Given two non-empty subsets A and B of the hyperbolic plane H2, we define their horocyclic Minkowski sum with parameter λ=1/2 as the set [A:B]1/2⊆H2 of all midpoints of horocycle curves connecting a point in A with a point in B. These horocycle curves are parameterized by hyperbolic arclength. The horocyclic Minkowski sum with parameter 0<λ<1 is defined analogously. We prove that when A and B are Borel-measurable,Area([A:B]λ)≥(1−λ)⋅Area(A)+λ⋅Area(B), where Area stands for hyperbolic area, with equality when A and B are concentric discs in the hyperbolic plane. We also give horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.

Volume preserving flows for convex curves and surfaces in the hyperbolic space

In this paper, we study the curvature flows of convex curves and surfaces in the hyperbolic space. In the first part, we consider the area preserving and length preserving κα-type curvature flows of smooth closed convex curves in the hyperbolic plane H2 and show that these two types of flows evolve convex curve to a geodesic circle exponentially in C∞ topology. In the second part, we study the volume preserving Gauss curvature flow of smooth closed convex surfaces in H3 and show that the solution remains convex and converges to a geodesic sphere exponentially in C∞ topology.

Convergence of elastic flows of curves into manifolds

For a given p∈[2,+∞), we define the p-elastic energy E of a closed curve γ:S1→M immersed in a complete Riemannian manifold (M,g) as the sum of the length of the curve and the Lp-norm of its curvature (with respect to the length measure). We are interested in the convergence of the (Lp,Lp′)-gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow.We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p=2 in the Euclidean space Rn, in the hyperbolic plane H2, and in the two-dimensional sphere S2. In particular, the result implies that such flow in Rn or H2 remains in a bounded region of the space for any time.

Fundamental domains in [formula omitted] for Fuchsian groups

In this paper, we provide a necessary and sufficient condition for a set in PSL(2,R) or in T1H2 to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane H2.

Hill’s equation, tire tracks and rolling cones

Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the ‘body cone’, along another, the ‘space cone’, with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean , thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular matrices in terms of rolling cones in 3-dimensional Minkowski space and the associated ‘pseudo spherical’ curves, in either the hyperbolic plane H2 or its Lorentzian analog H1,1. In particular, this yields an apparently new geometric interpretation of Schrödinger’s (or Hill’s) equation in terms of rolling without slipping of curves in the hyperbolic plane.

Real hyperbolic hyperplane complements in the complex hyperbolic plane

This paper studies Riemannian manifolds of the form M∖S, where M4 is a complete four dimensional Riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane CH2, and S is a compact totally geodesic codimension two submanifold whose induced Riemannian metric is modeled on the real hyperbolic plane H2. In this paper we write the metric on CH2 in polar coordinates about S, compute formulas for the components of the curvature tensor in terms of arbitrary warping functions (Theorem 7.1), and prove that there exist warping functions that yield a complete finite volume Riemannian metric on M∖S whose sectional curvature is bounded above by a negative constant (Theorem 1.1(1)). The cases of M∖S modeled on Hn∖Hn−2 and CHn∖CHn−1 were studied by Belegradek in [4] and [3], respectively. One may consider this work as “part 3” to this sequence of papers.

The geometry of Gaussian integer continued fractions

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship |ad−bc|=1 for a,b,c,d integers as it arises in the Farey tessellation of the hyperbolic plane H2 and the array of Ford circles in the upper-half of the real plane R2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition |αγ−βδ|=1 for α,β,γ,δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.

Computing the signatures of subgroups of non-Euclidean crystallographic groups

A (planar and cocompact) non-Euclidean crystallographic (NEC) group Δ is a subgroup of the group of, conformal and anti-conformal, isometries of the hyperbolic plane H2 such that H2/Δ is compact. NEC groups are classified algebraically by a symbol called signature. In this symbol there is a sign + or − and, in the case of sign +, some cycles of integers in the signature, called period-cycles, have an essential direction. In 1990 A.H.M. Hoare gave an algorithm to obtain the signature of a finite index subgroup of an NEC group. The process of Hoare fails in some cases in the task of computing the direction of period-cycles. In this work we complete the algorithm of Hoare, this allows us to construct a program for computing the signature of subgroups of NEC groups in all cases.

On decompositions of the hyperbolic plane satisfying many congruences

We generalize a theorem of J. F. Adams (‘On decompositions of the sphere’, J. Lond. Math. Soc. 29 (1954) 96–99). Then we apply the generalization to show that all large systems of congruences are solvable in the hyperbolic plane H2.

Complete surfaces in the product spaces Q ε 2 × S1 with constant Gaussian curvature

In the present paper, we first formulate and classify complete rotational surfaces with constant Gaussian curvature in the product spaces Qe2 × S1, where Qe2 denotes either the unit sphere S2 (when e = 1) or the hyperbolic plane H2 of constant curvature −1 (when e = −1). On the basis of this, we establish existence and uniqueness theorems for more general complete surfaces immersed in either S2 × S1 or in H2 ×S1. As the result, we established a classification for these surfaces in Qe2 ×S1 with a given constant K as the Gaussian curvature.

Side parameter for the torus with a single cone point

We introduce the side parameter map from the space of cone hyperbolic structures on the torus with a single cone point to the hyperbolic plane ℍ2, based on the idea in the Jorgensen theory for punctured torus Kleinian groups. Then, we show that the side parameter map is a homeomorphism.

A controlling norm for energy-critical Schrödinger maps

We consider energy-critical Schrödinger maps with target either the sphere S 2 \mathbb {S}^2 or hyperbolic plane H 2 \mathbb {H}^2 and establish that a unique solution may be continued so long as a certain space-time L 4 L^4 norm remains bounded. This reduces the large data global wellposedness problem to that of controlling this norm.

Simple continued fractions and cutting sequences

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ℍ2, the modular group Γ = PSL(2, ℤ) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ℝ and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T 0-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ℝ, the Farey tessellation and the simple continued fraction expansion of w ∈ ℝ ∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.

The anisotropic oscillator on the 2D sphere and the hyperbolic plane

An integrable generalization on the 2D sphere S2 and the hyperbolic plane H2 of the Euclidean anisotropic oscillator Hamiltonian with ‘centrifugal’ terms given byis presented. The resulting generalized Hamiltonian depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S2 (κ > 0), H2 (κ < 0) and E2 (κ = 0). Moreover, is explicitly shown to be integrable for any values of the parameters δ, Ω, λ1 and λ2. Therefore, can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω → 0 of . Furthermore, numerical integration of some of the trajectories for are worked out and the dynamical features arising from the introduction of a curved background are highlighted.The superintegrability issue for is discussed by focusing on the value Ω = 3δ, which is one of the cases for which the Euclidean Hamiltonian is known to be superintegrable (the 1 : 2 oscillator). We show numerically that for Ω = 3δ the curved Hamiltonian presents nonperiodic bounded trajectories, which seems to indicate that provides a non-superintegrable generalization of even for values of Ω that lead to commensurate frequencies in the Euclidean case. We compare this result with a previously known superintegrable curved analogue of the 1 : 2 Euclidean oscillator, which is described in detail, showing that the Ω = 3δ specialization of does not coincide with . Hence we conjecture that would be an integrable (but not superintegrable) curved generalization of the anisotropic oscillator that exists for any value of Ω and has constants of the motion that are quadratic in the momenta. Thus each commensurate Euclidean oscillator could admit another specific superintegrable curved Hamiltonian which would be different from and endowed with higher order integrals. Finally, the geometrical interpretation of the curved ‘centrifugal’ terms appearing in is also discussed in detail.

Isoperimetric inequalities for wave fronts and a generalization of Menzin's conjecture for bicycle monodromy on surfaces of constant curvature

Abstract The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length L bounding area A on a surface of constant curvature c, L 2 ≥ 4πA – cA 2 with equality holding only if the curve is a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss “bicycle curves” using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame), as discussed in [Finn, The College Mathematics Journal, 33, 2002], [Tabachnikov, Israel J. Math. 151: 1–28, 2006], and [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009]. We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H 2 and on the sphere S 2. We prove results along the lines of those in [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009] and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.

The Gauss Map of Minimal Surfaces in the Heisenberg Group

We study the Gauss map of minimal surfaces in the Heisenberg group endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane ℍ2. Conversely, any nowhere antiholomorphic harmonic map into ℍ2 is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.

The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these “curved” harmonic oscillators simultaneously on any such configuration space, using a Cayley–Klein (CK)-type approach, with two free parameters κ1,κ2 which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere S2, hyperbolic plane H2, AntiDeSitter sphere AdS1+1, and DeSitter sphere dS1+1) appear in this family, with Euclidean and Minkowski spaces as flat particular cases. We solve the equations of motion for the curved harmonic oscillator and obtain explicit expressions for the orbits by using three different methods: by direct integration, by obtaining the general CK version of Binet’s equation, and finally as a consequence of its superintegrable character. The orbits are conics with center at the potential origin on any CK space, thereby extending this well known Euclidean property to any constant curvature configuration space. The final part of the article, that has a more geometric character, presents pertinent results of the theory of conics on spaces of constant curvature.

The quantum harmonic oscillator on the sphere and the hyperbolic plane: κ-dependent formalism, polar coordinates, and hypergeometric functions

A nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates (r,ϕ) by making use of a curvature-dependent formalism. The curvature κ is considered as a parameter and then the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere S2 (κ&amp;gt;0) and the hyperbolic plane H2 (κ&amp;lt;0). A comparative study between the spherical and the hyperbolic quantum results is presented.