Constant Negative Curvature Research Articles

Special homogeneous surfaces

Abstract We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

Regularity with Respect to the Parameter of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting

We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in [ 0 , 1 ] [0,1] ), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are nonflat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complementary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.

Every noncompact surface is a leaf of a minimal foliation

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3 -manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C^{2} -smoothable.

A comprehensive thermodynamic study of the Reissner-Nordström-de sitter black hole with a constant negative curvature surrounded by a quintessence-like anisotropic fluid

The study investigates the thermodynamic characteristics of Reissner-Nordström black holes in the presence and absence of a quintessence-like fluid. We establish the relationships between key variables, such as mass, temperature, specific heat and Helmholtz free energy, in terms of entropy considering different scenarios. We also assess the value of the horizon radius for the second-order phase transition in the presence of a quintessence-like fluid. We also focus on the effects of logarithmic corrections in entropy, highlighting how these corrections cause the entropy ratio to deviate from unity. Moreover, we discuss the black hole’s equation of state, which serves as the foundation for deriving and plotting the temperature and P-V isotherm. In order to incorporate concepts like Planck’s length, we consider a generalized version of Heisenberg’s uncertainty principle and derive the constraints on the event horizon radius.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Dirac Landau levels for surfaces with constant negative curvature

Dirac Landau levels for surfaces with constant negative curvature

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

Anomalous and Chern topological waves in hyperbolic networks

Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior.

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant.

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Hyperbolic photonic topological insulators

Topological photonics provides a new degree of freedom to robustly control electromagnetic fields. To date, most of established topological states in photonics have been employed in Euclidean space. Motivated by unique properties of hyperbolic lattices, which are regular tessellations in non-Euclidean space with a constant negative curvature, the boundary-dominated hyperbolic topological states have been proposed. However, limited by highly crowded boundary resonators and complicated site couplings, the hyperbolic topological insulator has only been experimentally constructed in electric circuits. How to achieve hyperbolic photonic topological insulators is still an open question. Here, we report the experimental realization of hyperbolic photonic topological insulators using coupled ring resonators on silicon chips. Boundary-dominated one-way edge states with pseudospin-dependent propagation directions have been observed. Furthermore, the robustness of edge states in hyperbolic photonic topological insulators is also verified. Our findings have potential applications in the field of designing high-efficient topological photonic devices with enhanced boundary responses.

Uniform bounds for Kloosterman sums of half-integral weight with applications

Abstract Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x, m and n. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x, m and n. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p ⁢ ( n ) {p(n)} . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

Lieb–Thirring Inequalities on Manifolds with Constant Negative Curvature

Lieb–Thirring Inequalities on Manifolds with Constant Negative Curvature

Преобразование Бианки катушки Миндинга

The work is devoted to the study of the Bianchi transform for surfac­es of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Mining top, the Minding coil, the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuens surface and the Dinis surface. The study of surfaces of constant negative Gaussian curvature (pseudospheri­cal surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudo­spherical surfaces with the theory of networks, with the theory of solitons, with non-linear differential equations and sin-Gordon equations is estab­lished. The sin-Gordon equation plays an important role in modern phys­ics. Bianchi transformations make it possible to obtain new pseudospheri­cal surfaces from a given pseudospherical surface. The Bianchi transform for the Minding coil is constructed. Using a mathematical package, the Minding coil and its Bianchi transform are constructed.

Liouville theory and the Weil-Petersson geometry of moduli space

Liouville theory describes the dynamics of surfaces with constant negative curvature and can be used to study the Weil-Petersson geometry of the moduli space of Riemann surfaces. This leads to an efficient algorithm to compute the Weil-Petersson metric to arbitrary accuracy using Zamolodchikov’s recursion relation for conformal blocks. For example, we compute the metric on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}0,4 numerically to high accuracy by considering Liouville theory on a sphere with four punctures. We numerically compute the eigenvalues of the Weil-Petersson Laplacian, and find evidence that the obey the statistics of a random matrix in the Gaussian Orthogonal Ensemble.

A specific model of Hilbert geometry on the unit disc

AbstractA new metric on the open 2-dimensional unit disk is defined making it a geodesically complete metric space whose geodesic lines are precisely the Euclidean straight lines. Moreover, it is shown that the unit disk with this new metric is not isometric to any hyperbolic model of constant negative curvature, nor to any convex domain in $$\mathbb {R}^2$$ R 2 equipped with its Hilbert metric.

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature

The Dirac Equation on Metrics of Eguchi-Hanson Type II with Negative Constant Scalar Curvature

Holographic CFTs on AdSd× Sn and conformal defects

We consider (d+n+1)-dimensional solutions of Einstein gravity with constant negative curvature. Regular solutions of this type are expected to be dual to the ground states of (d + n)-dimensional holographic CFTs on AdSd × Sn. Their only dimensionless parameter is the ratio of radii of curvatures of AdSd and Sn. The same solutions may also be dual to (d − 1)-dimensional conformal defects in holographic QFTd+n. We solve the gravity equations with an associated conifold ansatz, and we classify all solutions both singular and regular by a combination of analytical and numerical techniques. There are no solutions, regular or singular, with two boundaries along the holographic direction. Out of the infinite class of regular solutions, only one is diffeomorphic to AdSd+n+1 and another to AdSd × AdSn+1. For the regular solutions, we compute the on-shell action as a function of the relevant parameters.