Constant Negative Curvature Research Articles

Computing hyperbolic choreographies

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, P\'{e}rez-Chavela and Reyes Victoria, we use stereographic projection and study the problem in the Poincar\'{e} disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in the companion paper. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.

Computing Quantum Bound States on Triply Punctured Two-Sphere Surface

We are interested in a quantum mechanical system on a triply punctured two-sphere surface with hyperbolic metric. The bound states on this system are described by the Maass cusp forms (MCFs) which are smooth square integrable eigenfunctions of the hyperbolic Laplacian. Their discrete eigenvalues and the MCF are not known analytically. We solve numerically using a modified Hejhal and Then algorithm, which is suitable to compute eigenvalues for a surface with more than one cusp. We report on the computational results of some lower-lying eigenvalues for the triply punctured surface as well as providing plots of the MCF using GridMathematica.

On small gaps in the length spectrum

We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrarily small gaps is topologically generic: this is established both for surfaces of constant negative curvature (Theorem 3.1) and for the space of negatively curved metrics (Theorem 4.1). While arbitrarily small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5.

A $$2\times 2$$ 2 × 2 Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets

We present a $$2\times 2$$2×2 Lax representation for discrete circular nets of constant negative Gaus curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Backlund transformations and an associated family. All the members of that family--although no longer circular--can be shown to have constant Gaus curvature as well. Explicit solutions for the Backlund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.

Strong submultiplicativity of the Poincaré metric

We give a direct proof of an important result of Solynin which says that the Poincare metric is a strongly submultiplicative domain function. This result is then used to define a new capacity for compact subsets of the complex plane $${\mathbb {C}}$$ , which might be called Poincare capacity. If the compact set $$K \subseteq {\mathbb {C}}$$ is connected, then the Poincare capacity of K is the same as the logarithmic capacity of K. In this special case, the submultiplicativity is well-known and can be stated as an inequality for the normalized conformal map onto the complement of K. Using the connection between Poincare metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction.

New improved massive gravity

We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first-order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space AdS3) and constant torsion provide exact solutions.

Resonances and density bounds for convex co-compact congruence subgroups of SL 2(ℤ)

Let Γ be a convex co-compact subgroup of SL2(Z), and let Γ(q) be the sequence of “congruence” subgroups of Γ. Let Rq ⊂ C be the resonances of the hyperbolic Laplacian on the “congruence” surfaces Γ(q)H2. We prove two results on the density of resonances in Rq as q → ∞: the first shows at least Cq3 resonances in slowly growing discs, the other one is a bound from above in boxes {δ/2 0 for all σ > δ/2, j = 1, 2. These two estimates highlight the role of the critical line \(\left\{ {\operatorname{Re} \left( s \right) = \frac{\delta }{2}} \right\}\) when looking at congruence families.

Cosmological wormholes

We describe in details the procedure how the Lobachevsky space can be factorized to a space of the constant negative curvature filled with a gas of wormholes. We show that such wormholes have throat sections in the form of tori and are traversable and stable in the cosmological context. The relation of such wormholes to the dark matter phenomenon is briefly described. We also discuss the possibility of the existence of analogous factorizations for all types of homogeneous spaces.

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ℍ 2n+1 which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ℍ 2n+1.

Finiteness of polygonal relative equilibria for generalised quasi-homogeneous n-body problems and n-body problems in spaces of constant curvature

We prove for generalisations of quasi-homogeneous n-body problems with centre of mass zero and n-body problems in spaces of negative constant Gaussian curvature that if the masses and rotation are fixed, there exists, for every order of the masses, at most one equivalence class of relative equilibria for which the point masses lie on a circle, as well as that there exists, for every order of the masses, at most one equivalence class of relative equilibria for which all but one of the point masses lie on a circle and rotate around the remaining point mass. The method of proof is a generalised version of a proof by J.M. Cors, G.R. Hall and G.E. Roberts on the uniqueness of co-circular central configurations for power-law potentials.

Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making.

Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.

Sigma models with negative curvature

We construct Higgs Effective Field Theory (HEFT) based on the scalar manifold H^n, which is a hyperbolic space of constant negative curvature. The Lagrangian has a non-compact O(n,1) global symmetry group, but it gives a unitary theory as long as only a compact subgroup of the global symmetry is gauged. Whether the HEFT manifold has positive or negative curvature can be tested by measuring the S-parameter, and the cross sections for longitudinal gauge boson and Higgs boson scattering, since the curvature (including its sign) determines deviations from Standard Model values.

Lobachevsky crystallography made real through carbon pseudospheres

We realize Lobachevsky geometry in a simulation lab, by producing a carbon-based energetically stable molecular structure, arranged in the shape of a Beltrami pseudosphere. We find that this structure: (i) corresponds to a non-Euclidean crystallographic group, namely a loxodromic subgroup of ; (ii) has an unavoidable singular boundary, that we fully take into account. Our approach, substantiated by extensive numerical simulations of Beltrami pseudospheres of different size, might be applied to other surfaces of constant negative Gaussian curvature, and points to a general procedure to generate them. Our results also pave the way to test certain scenarios of the physics of curved spacetimes owing to graphene’s unique properties.

On the sub-Riemannian geometry of contact Anosov flows

We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

Stable Degeneracies for Ising Models

We introduce and consider the notion of stable degeneracies of translation invariant energy functions for finite Ising models. By this term we mean the lack of injectivity that cannot be lifted by changing the interaction. We show that besides the symmetry-induced degeneracies, related to spin flip, translation and reflection, there exist additional stable degeneracies, due to more subtle symmetries. One such symmetry is the one of the Singer group of a finite projective plane. Others are described by combinatorial relations akin to trace identities. Our results resemble traits of the length spectrum for closed geodesics on a Riemannian surface of constant negative curvature. There stable degeneracy is defined w.r.t. Teichm\"uller space as parameter space.

Long-Time Dynamics of the Perturbed Schrödinger Equation on Negatively Curved Surfaces

We consider perturbations of the semiclassical Schrodinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the perturbation, the solutions associated to initial data in a small spectral window become equidistributed in the semiclassical limit. As an application of our method, we also derive some properties of the quantum Loschmidt echo below and beyond the Ehrenfest time for initial data in a small spectral window.

Complex and quaternionic hyperbolic Kleinian groups with real trace fields

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $\Gamma$ is irreducible, $\Gamma$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.

Brownian motion on treebolic space: positive harmonic functions

Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T with vertex degree p+1 < 2, the latter seen as a one-complex. In a previous paper [arXiv:1212.6151, Rev. Mat. Iberoamericana, in print] we have explored the metric structure and isometry group of that space. Relying on the analysis on strip complexes, a family of natural Laplacians with "vertical drift" and the escape to infinity of the associated Brownian motion were considered. Here, we undertake a potential theoretic study, investigating the positive harmonic functions associated with those Laplacians. The methodological subtleties stem from the singularites of treebolic space at its bifurcation lines. We first study harmonic functions on simply connected sets with "rectangular" shape that are unions of strips. We derive a Poisson representation and obtain a solution of the Dirichlet problem on sets of that type. This provides properties of the density of the induced random walk on the collection of all bifuraction lines. Subsequently, we prove that each positive harmonic function with respect to that random walk has a unique extension which is harmonic with respect to the Laplacian on treebolic space. Finally, we derive a decomposition theorem for positive harmonic functions on the entire space that leads to a characterisation of the weak Liouville property. We determine all minimal harmonic functions in those cases where our Laplacian arises from lifting a (smooth) hyperbolic Laplacian with drift from the hyperbolic plane to treebolic space.

Hanbury Brown and Twiss measurements in curved space

Hanbury Brown and Twiss measurements on speckle patterns propagating along curved surfaces improve understanding of spatial coherence. When Hanbury Brown and Twiss (HBT) proposed their technique of intensity correlation measurements1,2,3 to examine the angular size of stars in the visible range, they challenged the common conception of quantum mechanics and kicked off a discussion that led to the establishment of quantum optics4,5,6. In this Letter we revisit this fundamental technique and study its implications in the presence of space curvature. To this end we theoretically and experimentally investigate the evolution of speckle patterns propagating along two-dimensional surfaces of constant positive and negative Gaussian curvature, defying the notion that light always gains spatial coherence during free-space propagation. We also discuss the measurability of the traversed space's curvature utilizing HBT from an inhabitant's point of view. Through their symmetry, surfaces with constant Gaussian curvature act as analogue models for universes possessing non-vanishing cosmological constants.