Surfaces Of Constant Negative Curvature Research Articles

Dimension matters when modeling network communities in hyperbolic spaces.

Over the last decade, random hyperbolic graphs have proved successful in providing geometric explanations for many key properties of real-world networks, including strong clustering, high navigability, and heterogeneous degree distributions. These properties are ubiquitous in systems as varied as the internet, transportation, brain or epidemic networks, which are thus unified under the hyperbolic network interpretation on a surface of constant negative curvature. Although a few studies have shown that hyperbolic models can generate community structures, another salient feature observed in real networks, we argue that the current models are overlooking the choice of the latent space dimensionality that is required to adequately represent clustered networked data. We show that there is an important qualitative difference between the lowest-dimensional model and its higher-dimensional counterparts with respect to how similarity between nodes restricts connection probabilities. Since more dimensions also increase the number of nearest neighbors for angular clusters representing communities, considering only one more dimension allows us to generate more realistic and diverse community structures.

Low temperature quantum bounds on simple models

In the past few years, there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity) and chaos (Lyapunov exponent), relevant at low temperatures. The interest comes from the fact that Black-Hole models seem to saturate all of them. The goal of this work is to gain physical intuition about the quantum mechanisms that enforce these bounds on simple models. To this aim, we consider classical and quantum free dynamics on curved manifolds. These systems exhibit chaos up to the lowest temperatures and -- as we discuss -- they violate the bounds in the classical limit. First of all, we show that the quantum dimensionless viscosity and the Lyapunov exponent only depend on the de Broglie length and a geometric length-scale, thus establishing the scale at which quantum effects become relevant. Then, we focus on the bound on the Lyapunov exponent and identify three different ways in which quantum effects arise in practice. We illustrate our findings on a toy model given by the surface of constant negative curvature — a paradigmatic model of quantum chaos — glued to a cylinder. By exact solution and numerical investigations, we show how the chaotic behaviour is limited by the quantum effects of the curvature itself. Interestingly, we find that at the lowest energies the bound to chaos is dominated by the longest length scales, and it is therefore a collective effect.

Errata and Addenda to: “Hydrodynamic Vortex on Surfaces” and “The Motion of a Vortex on a Closed Surface of Constant Negative Curvature”

The two papers in the title contain some equations that are not complete. The missing terms, which are of topological origin, were recently unveiled by Björn Gustafsson. In this note, we present the equations of Gustafsson in the case of a single vortex in a compact boundaryless surface, and show that many conclusions we have reached with the incomplete equations remain valid. It seems that the extra-terms in Gustafsson’s equations can be explicitly written in terms of elementary functions only for the two-torus, and this is done at the end of this note.

Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups

Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichm\"uller space, taking all values ("flexibility") between zero and a maximum, which is achieved on the surface that admits a regular fundamental $(8g-4)$-gon. We obtain explicit formulas for both entropies. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof -- valid only for certain multi-parameters -- uses the realization of geodesic flow as a special flow over the natural extension of the boundary map.

Control of eigenfunctions on surfaces of variable curvature

We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.

Extremal parameters and their duals for boundary maps associated to Fuchsian groups

We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen–Series boundary maps, and their natural extensions, associated to co-compact torsion-free Fuchsian groups. If the boundary map parameters are extremal—that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon—then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. Moreover, each extremal parameter choice has a corresponding dual parameter choice such that the “past” of the arithmetic code of a geodesic is the “future” for the code using the dual parameter. This duality was observed for two classical parameter choices by Adler and Flatto; here we show constructively that every extremal parameter choice has a dual.

Exploring event horizons and Hawking radiation through deformed graphene membranes

Analogue gravitational systems are becoming an increasing popular way of studying the behaviour of quantum systems in curved spacetime. Setups based on ultracold quantum gases in particular, have been recently harnessed to explore the thermal nature of Hawking’s and Unruh’s radiation that was theoretically predicted almost 50 years ago. For solid state implementations, a promising system is graphene, in which a link between the Dirac-like low-energy electronic excitations and relativistic quantum field theories has been unveiled soon after its discovery. This link could be extended to the case of curved quantum field theory when the graphene sheet is shaped in a surface of constant negative curvature, known as Beltrami’s pseudosphere. Here we provide numerical evidence that energetically stable negative curvature graphene surfaces can be realized. Owing to large-scale simulations, our geometrical realizations are characterized by a ratio between the carbon–carbon bond length and the pseudosphere radius small enough to allow the formation of an analog of a black hole event horizon. Additionally, from the energy dependence of the spatially resolved density of states, we infer some thermal properties of the corresponding gravitational system, which could be investigated using low temperature scanning tunnelling microscopy or optical near field spectroscopy. These findings pave the way to the realization of a solid-state system in which the curved spacetime dynamics of quantum many body systems can be investigated.

A solution to Flinn’s conjecture on weakly expansive flows

In L. W. Flinn’s PhD thesis published in 1972, the author conjectured that weakly expansive flows are also expansive flows. In this paper we use the horocycle flow on compact Riemann surfaces of constant negative curvature to show that Flinn’s conjecture is not true.

Nonorthogonality constraints in open quantum and wave systems

It is known that the overlap of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an appropriately defined decay operator. Here, we show that the weaker and more realistic condition of positive semi-definiteness is sufficient. We prove also that the bound becomes an equality for the case of single-channel decay. However, we show that the condition of positive semi-definiteness can be spoiled by quantum backflow. Hence, the overlap of quasibound quantum states subjected to outgoing-wave conditions can be larger than expected from the bound. A modified and less stringent bound, however, can be introduced. For electromagnetic systems, it turns out that a modification of the bound is not necessary due to the linear free-space dispersion relation. Finally, a geometric interpretation of the nonorthogonality bound is given which reveals that in this context the complex energy space can seen as a surface of constant negative curvature.

Expansiveness for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature

We study expansive properties for the geodesic and horocycle flows on compact Riemann surfaces of constant negative curvature. It is well-known that the geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow is positive and negative separating in the sense of Gura. In this paper, we give a new proof of the expansiveness in the sense of Bowen-Walters for the geodesic flow and show that the horocycle flow is positive and negative kinematic expensive in the sense of Artigue as well as expansive in the sense of Katok-Hasselblatt but not expensive in the sense of Bowen-Walters. We also point out that the geodesic flow is neither positive nor negative separating.

Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature

Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature

Hyperbolic relative equilibria for the negative curved n–body problem

We consider the n–body problem defined on surfaces of constant negative curvature. For the five and seven body case with a symmetric configuration, we give conditions on initial positions for the existence of collinear hyperbolic relative equilibria, where here collinear means that the bodies are on the same geodesic. If they satisfy the above conditions, then we give explicitly the values of the masses in terms of the positions, such that they can lead to these type of relative equilibria. The set of parameters that lead to these type of solutions has positive Lebesgue measure. For the general case of n–equal masses, we prove the no-existence of hyperbolic relative equilibria with a regular polygonal shape. In particular the Lagrangian hyperbolic relative equilibria do not exist.

The motion of a vortex on a closed surface of constant negative curvature.

The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace-Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolza surface. This is the first determination of the orbits of a vortex on a closed surface of genus greater than one. The numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the Bolza surface. Some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc. The following theorem is proved: 'a Weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium'.

Spectral properties of horocycle flows for compact surfaces of constant negative curvature

We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wuflows and we show that Wu flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C2. This generalises recent results on time changes ofhorocycle flows.

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.

Hypoelliptic Laplacian and probability

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

Analytic vortex solutions on compact hyperbolic surfaces

We construct, for the first time, abelian Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations.

Eisenstein Quasimodes and QUE

We consider the question of quantum unique ergodicity (QUE) for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series on \({SL(2,\mathbb{Z})\backslash\mathbb{H}}\) , extending results of Luo–Sarnak and Jakobson. Moreover, we observe that the equidistribution results of Luo–Sarnak and Jakobson extend to quasimodes of much weaker order—for which QUE is known to fail on compact surfaces—though in this scenario the total mass of the limit measures will decrease. We interpret this stronger equidistribution property in the context of arithmetic QUE, in light of recent joint work with Lindenstrauss (Invent Math 198(1), 219–259, 2014) on joint quasimodes.

Constant Gaussian curvature surfaces in the 3-sphere via loop groups

In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in $S^3$ with Gauss curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if $K$ is constant. We give a uniform loop group formulation for all such surfaces with $K\neq 0$, and use the generalized d'Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with $K<0$ and those with $0<K<1$.

Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that

The solutions of many issues, of the ongoing efforts to make deformed graphene a tabletop quantum field theory in curved spacetimes, are presented. A detailed explanation of the special features of curved spacetimes, originating from embedding portions of the Lobachevsky plane into $\mathbf{R}^3$, is given, and the special role of coordinates for the physical realizations in graphene, is explicitly shown, in general, and for various examples. The Rindler spacetime is reobtained, with new important differences with respect to earlier results. The de Sitter spacetime naturally emerges, for the first time, paving the way to future applications in cosmology. The role of the BTZ black hole is also briefly addressed. The singular boundary of the pseudospheres, "Hilbert horizon", is seen to be closely related to event horizon of the Rindler, de Sitter, and BTZ kind. This gives new, and stronger, arguments for the Hawking phenomenon to take place. An important geometric parameter, $c$, overlooked in earlier work, takes here its place for physical applications, and it is shown to be related to graphene's lattice spacing, $\ell$. It is shown that all surfaces of constant negative curvature, ${\cal K} = -r^{-2}$, are unified, in the limit $c/r \to 0$, where they are locally applicable to the Beltrami pseudosphere. This, and $c = \ell$, allow us a) to have a phenomenological control on the reaching of the horizon; b) to use spacetimes different than Rindler for the Hawking phenomenon; c) to approach the generic surface of the family. An improved expression for the thermal LDOS is obtained. A non-thermal term for the total LDOS is found. It takes into account: a) the peculiarities of the graphene-based Rindler spacetime; b) the finiteness of a laboratory surface; c) the optimal use of the Minkowski quantum vacuum, through the choice of this Minkowski-static boundary.