Three-dimensional Hyperbolic Space Research Articles

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]‒1) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text]‒1 immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text]‒1 surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

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.

On the asymptotic geometry of finite-type $k$-surfaces in three-dimensional hyperbolic space

For 0<k<1 , a finite-type k - surface in 3 -dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k . In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k -surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Möbius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k -surfaces. The first, which will be called the generalized volume , is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy , is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k -surfaces by open complex manifolds, we prove a new Schläfli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Möbius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schläfli-type formula.

Conformal capacity of hedgehogs

We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets E E in the unit disk D = { z : | z | > 1 } \mathbb {D}=\{z:\,|z|>1\} consisting of a central body E 0 E_0 that is typically a smaller disk D ¯ r = { z : | z | ≤ r } \overline {\mathbb {D}}_r=\{z:\,|z|\le r\} , 0 > r > 1 0>r>1 , and several spikes E k E_k that are compact sets lying on radial intervals I ( α k ) = { t e i α k : 0 ≤ t > 1 } I(\alpha _k)=\{te^{i\alpha _k}:\,0\le t>1\} . The main questions we are concerned with are the following: (1) How does the conformal capacity c a p ( E ) \mathrm {cap}(E) of E = ∪ k = 0 n E k E=\cup _{k=0}^n E_k behave when the spikes E k E_k , k = 1 k=1 , …, n n , move along the intervals I ( α k ) I(\alpha _k) toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity c a p ( E ) \mathrm {cap}(E) depend on the distribution of angles between the spikes E k E_k ? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.

The abstract Birman—Schwinger principle and spectral stability

We discuss abstract Birman—Schwinger principles to study spectra of self-adjoint operators subject to small non-self-adjoint perturbations in a factorised form. In particular, we extend and in part improve a classical result by Kato which ensures that the spectrum does not change under small perturbations. As an application, we revisit known results for Schrödinger and Dirac operators in Euclidean spaces and establish new results for Schrödinger operators in three-dimensional hyperbolic space.

Symplectic groups over noncommutative algebras

We introduce the symplectic group \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) over a noncommutative algebra A with an anti-involution \(\sigma \). We realize several classical Lie groups as \({{\,\mathrm{Sp}\,}}_2\) over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space \(X_{{{\,\mathrm{Sp}\,}}_2(A,\sigma )}\), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\)) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.

Boundary metrics on soliton moduli spaces

The geodesic approximation is a powerful method for studying the dynamics of BPS solitons. However, there are systems, such as BPS monopoles in three-dimensional hyperbolic space, where this approach is not applicable because the moduli space metric defined by the kinetic energy is not finite. In the case of hyperbolic monopoles, an alternative metric has been defined using the abelian connection on the sphere at infinity, but its relation to the dynamics of hyperbolic monopoles is unclear. Here this metric is placed in a more general context of boundary metrics on soliton moduli spaces. Examples are studied in systems in one and two space dimensions, where it is much easier to compare the results with simulations of the full nonlinear field theory dynamics. It is found that geodesics of the boundary metric provide a reasonable description of soliton dynamics.

Geometry and entanglement in the scattering matrix

A formulation of nucleon–nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix –that appears at leading-order in the EFT description– possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.

Semiclassical p-branes in hyperbolic space

The one-loop effects to the Dirac action of p-branes in a hyperbolic background from the path integral and the solution of the Wheeler–DeWitt equation are analysed. The objective of comparing the equivalent quantization procedures is to study in detail the validity of the semiclassical approximation and divergences associated to one-loop corrections. This is in line with a bottom-up approach to holographic Wilson loops. We employ the heat kernel regularization method for both quantization procedures and we study in great detail one-loop corrections to geodesics in a two-dimensional hyperbolic space and semi-spheres in a three-dimensional hyperbolic space. We show that the divergences, given by the high energy expansion of the heat kernel, can be classified by their compatibility with the semiclassical approximation and geometric nature.

Prime geodesic theorem for the Picard manifold

Let Γ=PSL(2,Z[i]) be the Picard group and H3 be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient Γ∖H3, called the Picard manifold, obtaining an error term of size O(X3/2+θ/2+ϵ), where θ denotes a subconvexity exponent for quadratic Dirichlet L-functions defined over Gaussian integers.

Critical properties of the Ising model in hyperbolic space.

The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two- and three-dimensional hyperbolic spaces using Monte Carlo and high- and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the "asymptotic" limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three-dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.

Hyperbolic geometry of the olfactory space.

In the natural environment, the sense of smell, or olfaction, serves to detect toxins and judge nutritional content by taking advantage of the associations between compounds as they are created in biochemical reactions. This suggests that the nervous system can classify odors based on statistics of their co-occurrence within natural mixtures rather than from the chemical structures of the ligands themselves. We show that this statistical perspective makes it possible to map odors to points in a hyperbolic space. Hyperbolic coordinates have a long but often underappreciated history of relevance to biology. For example, these coordinates approximate the distance between species computed along dendrograms and, more generally, between points within hierarchical tree-like networks. We find that both natural odors and human perceptual descriptions of smells can be described using a three-dimensional hyperbolic space. This match in geometries can avoid distortions that would otherwise arise when mapping odors to perception.

Singularities of worldsheets in spherical space–times

In this paper, the singularities of the geometry for four classes of worldsheets, which are respectively, located in three-dimensional hyperbolic space and three-dimensional de Sitter space–time are considered. Under the theoretical frame of geometry of space–time and as applications of singularity theory, it is shown that these worldsheets have two classes of singularities, that is, in the local sense, these four classes of worldsheets are, respectively, diffeomorphic to the cuspidal edge and the swallowtail. The first hyperbolic worldsheet and the second hyperbolic worldsheet are [Formula: see text]-dual to the tangent curves of spacelike curves. Moreover, it is also revealed that there is a close relationship between the types of singularities of worldsheets and a geometric invariant [Formula: see text], depending on whether [Formula: see text] or [Formula: see text] and [Formula: see text], the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.

A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.

Combinatorial Decompositions, Kirillov–Reshetikhin Invariants, and the Volume Conjecture for Hyperbolic Polyhedra

ABSTRACTWe suggest a method of computing volume for a simple polytope P in three-dimensional hyperbolic space . This method combines the combinatorial reduction of P as a trivalent graph Γ (the 1-skeleton of P) by I–H, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalized tetrahedra. With each decomposition (under some conditions), we associate a potential function Φ such that the volume of P can be expressed through a critical values of Φ. The results of our numeric experiments with this method suggest that one may associate the above-mentioned sequence of combinatorial moves with the sequence of moves required for computing the Kirillov–Reshetikhin invariants of the trivalent graph Γ. Then the corresponding geometric decomposition of P might be used in order to establish a link between the volume of P and the asymptotic behavior of the Kirillov–Reshetikhin invariants of Γ, which is colloquially known as the Volume Conjecture.

The Parabola Theorem on Continued Fractions

Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Mobius transformations. This geometric approach allows us to relate the Stern–Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Mobius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions.

Profile decompositions for wave equations on hyperbolic space with applications

The goal for this paper is twofold. Our first main objective is to develop Bahouri–Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as $$t \rightarrow \pm \infty $$ . This proof serves as a blueprint for the arguments in the work [45] by the authors, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

Yang–Mills moduli space in the adiabatic limit

We consider the Yang–Mills equations for a matrix gauge group G inside the future light cone of four-dimensional Minkowski space, which can be viewed as a Lorentzian cone over the three-dimensional hyperbolic space H3. Using the conformal equivalence of and the cylinder we show that, in the adiabatic limit when the metric on H3 is scaled down, classical Yang–Mills dynamics is described by geodesic motion in the infinite-dimensional group manifold of smooth maps from the boundary two-sphere into the gauge group G.

Rectifying Curves in the Three-Dimensional Hyperbolic Space

B. Y. Chen introduced rectifying curves in \({\mathbb{R}^3}\) as space curves whose position vector always lies in its rectifying plane. Recently, the authors have extended this definition (as well as several results about rectifying curves) to curves in the three-dimensional sphere. In this paper, we study rectifying curves in the three-dimensional hyperbolic space, and obtain some results of characterization and classification for such kind of curves. Our results give interesting and significant differences between hyperbolic, spherical and Euclidean geometries.

FRACTALS WITH HYPERBOLIC SCATORS IN 1 + 2 DIMENSIONS

A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled [Formula: see text](s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.