Real Hyperbolic Space Research Articles

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces C H n \mathbb C H^n , n ≥ 3 n \geq 3 . For the quaternionic hyperbolic spaces H H n \mathbb H H^n , n ≥ 3 n \geq 3 , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

Geometry and dynamics of discrete isometry groups of higher rank symmetric spaces

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of “nonelementary” discrete isometry groups of higher rank symmetric spaces. We obtain the natural generalisations of some well-known results from Kleinian group theory. Our main tool consists in a detailed description of the dynamics of individual isometries. As a by-product, we give a new geometric construction of free isometry groups with parabolic elements in higher rank symmetric spaces.

Bohr phenomena for Laplace-Beltrami operators

We investigate a Bohr phenomenon on the spaces of solutions of weighted Laplace-Beltrami operators associated with the hyperbolic metric of the unit ball in ℂ N . These solutions do not satisfy the usual maximum principle, and the spaces have natural bases none of whose members is a constant function. We show that these bases exhibit a Bohr phenomenon, define a Bohr radius for them that extends the classical Bohr radius, and compute it exactly. We also compute the classical Bohr radius of the invariant harmonic functions on the real hyperbolic space.

Poisson kernels of half–spaces in real hyperbolic spaces

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space \mathbb{H}^n . This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space \mathbb{H}^n . For n=3 , 4 or 6 we compute explicit formulas for the Poisson kernel itself.

Non-Arithmetic Uniformization of Some Real Moduli Spaces

Some real moduli spaces can be presented as real hyperbolic space modulo a non-arithmetic group. The whole moduli space is made from some incommensurable arithmetic pieces, in the spirit of the construction of Gromov and Piatetski-Shapiro.

Fluxes, brane charges and Chern morphisms of hyperbolic geometry

The purpose of this paper is to provide the reader with a collection of results which can be found in the mathematical literature and to apply them to hyperbolic spaces that may have a role in physical theories. Specifically, we apply K-theory methods for the calculation of brane charges and RR-fields on hyperbolic spaces (and orbifolds thereof). It is known that by tensoring K-groups with the rationals, K-theory can be mapped to rational cohomology by means of the Chern character isomorphisms. The Chern character allows one to relate the analytic Dirac index to a topological index, which can be expressed in terms of cohomological characteristic classes. We obtain explicit formulae for Chern character, spectral invariants, and the index of a twisted Dirac operator associated with real hyperbolic spaces. Some notes for a bivariant version of topological K-theory (KK-theory) with its connection to the index of the twisted Dirac operator and twisted cohomology of hyperbolic spaces are given. Finally, we concentrate on lower K-groups useful for description of torsion charges.

Brane charges and Chern–Simons invariants of hyperbolic spaces, with cosmological applications

We discuss methods of K-theory associated with hyperbolic orbifolds and valid for the description of Chern morphisms and brane charges. Such methods of K-theory are applied to compute D-brane charges, which are identified with elements of Grothendick K-groups, and for manifolds with horizons, spaces that naturally arise as the near-horizon of black brane geometries. In de Sitter spaces, these solutions break supersymmetry, and do not describe universes with zero cosmological constant. Here we pay attention to real hyperbolic spaces, and we examine associated Chern classes and brane charges using methods of K-theory and spectral theory of differential operators related to real hyperbolic spaces. An argument in favour of hyperbolic geometries in the treatment of the contributions to the vacuum persistence amplitude in QFT is given. All those are to be viewed as the proper mathematical structures underlying QFT with relevant backgrounds and boundary conditions in string cosmology.

On Sampling Formulas on Symmetric Spaces

We give a sampling formula using the Radon transform along a maximal geodesic subspace of the Riemannian symmetric space. For the real hyperbolic space we can get a total sampling formula. To get this formula, we prepare a sampling formula for the sphere.

Hitting distributions of geometric Brownian motion

Letbe the first hitting time of the point 1 by the geometric Brownian motion X(t) = xexp(B(t) −2µt) with drift µ > 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with E 0 B 2 (t) = 2t. We provide an integral formula for the density function of the stopped exponential functional A(�) = R � 0 X 2 (t)dt and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in (BGS), the present paper also covers the case of arbitrary drifts µ > 0 and provides a significant unification and extension of results of the above-mentioned paper. As a corollary we provide an integral formula and give asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.

THERMODYNAMICS OF ABELIAN GAUGE FIELDS IN REAL HYPERBOLIC SPACES

We work with N-dimensional compact real hyperbolic space XΓ with universal covering M and fundamental group Γ. Therefore, M is the symmetric space G/K, where G = SO 1(N, 1) and K = SO (N) is a maximal compact subgroup of G. We regard Γ as a discrete subgroup of G acting isometrically on M, and we take XΓ to be the quotient space by that action: XΓ = Γ∖M = Γ∖G/K. The natural Riemannian structure on M (therefore on X) induced by the Killing form of G gives rise to a connection p-form Laplacian 𝔏p on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on Abelian p-forms on the real compact hyperbolic manifold XΓ. The spectral zeta function related to the operator 𝔏p, considering only the coexact part of the p-forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic functions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.

Orbifold compactification and solutions of M-theory from Milne spaces

In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications $SL(2,{\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$, where $Id$ is the identity matrix, is analyzed in detail. The spectrum of a eleven-dimensional field theory can be obtained with the help of the theory of harmonic functions in the fundamental domain of this group and it is associated with the cusp forms and the Eisenstein series. The supersymmetry surviving for supergravity solutions involving real hyperbolic space factors is briefly discussed.

Poisson transformation for one-sheeted hyperboloids

This paper is concerned with the study of the Poisson transformation and with finding the complete asymptotic series at infinity for this transformation (not just its dominant term) for an important subclass of the class of semisimple symmetric spaces with non-compact : the real hyperbolic spaces (one-sheeted hyperboloids) . The expansions are obtained for arbitrary (not necessarily -finite) functions. The operators involved in the coefficients of the expansions and acting on functions at the boundary are described. The reducibility and the irreducibility of eigensubspaces of the Laplace-Beltrami operator in function spaces on the hyperboloid are studied. The structure of representations acting in spaces of eigenfunctions is described. Descriptions of the kernel and the closure of the range of the Poisson transformation are presented.

Isometric actions of Lie subgroups of the Moebius group

We prove here, by geometric, or rather dynamical, methods, the following theorem. Let G be a non-compact connected Lie subgroup of the isometry group of the real hyperbolic space , which does not fix any point at infinity, i.e. on . Then G preserves a certain hyperbolic subspace and ‘contains’ all the identity components of its isometry group. We provide an ‘algebra-free’ proof and present the dynamical tools used, so that the exposition is ‘self-contained’.

La Couronne d'un Cône Symétrique

The crown of a Riemannian symmetric space of non compact type is a domain in its complexification with the following remarkable property: All eigenfunctions of the invariant differential operators have a holomorphic extension to the crown, and this domain is maximal for this property. We explain how it follows from recent results by Krotz and Stanton, and by Barchini. Furthermore we describe the geometry of the crown of a symmetric cone, and finally we give several realizations of the crown of the real hyperbolic space.

Hardy's theorem for the Jacobi transform

Let ab 1⁄4 1 4 for positive constants a; b. Hardy’s theorem states that the function f ðxÞ 1⁄4 e ax is the only function (modulo constants) satisfying the decay conditions f ðxÞ 1⁄4 Oðe ax Þ and f ðxÞ 1⁄4 Oðe bx Þ, where f denotes the Fourier transform of f . We generalise this theorem and its L p analogues to the Jacobi transform. We then consider the Fourier transform on the real hyperbolic spaces SOoðm; nÞ=SOoðm 1; nÞ, m; n A N, and show, as an application of our results for the Jacobi transform, that Hardy’s theorem only can be generalised to the Riemannian ðm 1⁄4 1Þ case. It can, in particular, not be generalised to SLð2;RÞFSUð1; 1ÞF SOoð2; 2Þ=SOoð1; 2Þ.

Harmonic morphisms from even-dimensional hyperbolic spaces

In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

Real hypersurfaces in nonflat complex space forms are irreducible

The study of real hypersurfaces in complex projective space and complex hyperbolic space has been an active field over the past three decades. Although these ambient spaces might be regarded as the simplest after the spaces of constant curvature, they impose significant restrictions on the geometry of their hypersurfaces. For instance, they do not admit totally umbilical hypersurfaces and Einstein hypersurfaces. On the other hand, several important classes of real hypersurfaces in complex projective space have been constructed and investigated by many geometers. For instance, H.B. Lawson investigated real hypersurfaces of constructed by Clifford minimal hypersurfaces of +1 via Hopf fibration. R. Takagi [9] gave the list of homogeneous real hypersurfaces of . Many geometers then study the geometry from the list of Takagi and obtained various interesting geometric characterizations of homogeneous real hypersurfaces in . Another important class of real hypersurfaces in which contains the list of R. Takagi is the class of Hopf hypersurfaces. Such hypersurfaces are real hypersurfaces whose structure vector ξ is a principal curvature vector, where is the complex structure and ξ is the unit normal vector field. Examples and geometric characterizations of Hopf hypersurfaces have also been obtained by various geometers. It is known that in , is a homogeneous real hypersurface if and only if is a Hopf hypersurface with constant principal curvatures [6, 9]. The study of real hypersurfaces in complex hyperbolic space has followed developments in , often with similar results, but sometimes with differences (see [1, 7, 8] for more details). It is well-known that real projective space and real hyperbolic space admit ample hypersurfaces which are the Riemannian products of some Riemannian manifolds. It is also well-known that 3 admits a complex hypersurface which is the Riemannian

$L^p$ versions of Hardy’s uncertainty principle on hyperbolic spaces

Hardy's uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove L P versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces.

Real hyperbolic on the outside, complex hyperbolic on the inside

The rank one symmetric spaces of negative curvature come in three infinite families: real hyperbolic space H; complex hyperbolic space CH; and quaternionic hyperbolic space QH. (The Cayley plane is the remaining example.) Aside from the obvious embeddings H ↪→ CH ↪→ QH the three geometries seem fairly unrelated to each other. For instance, H admits non-arithmetic lattices in all dimensions [GrP] while QH only admits arithmetic lattices [GrS]. (See [C] for a related result.) The question of non-arithmetic lattices in CH is a basic unsolved problem [DM]. For a representation-theoretic comparison of discrete subgroups in the different rank one spaces, see [Sh]. In this paper we make a new connection between H and CH. We construct a closed hyperbolic 3-manifold which (as a diffeomorphic copy) is the ideal boundary of a complex hyperbolic 4-manifold. Up to index 2, the isometry group of CH is PU(2, 1), the group of complex projective automorphisms of the unit ball in C. The ideal boundary of CH is the unit 3-sphere S. A spherical CR structure on a 3-manifold is a system of coordinate charts into S whose transition functions are restrictions of elements of PU(2, 1). While plenty of closed Seifert fibered manifolds admit spherical CR structures [KT], our example gives the only known spherical CR structure on a closed hyperbolic 3-manifold.

Radon, Cosine and Sine Transforms on Real Hyperbolic Space

New pointwise inversion formulae are obtained for the d-dimensional totally geodesic Radon transform on the n-dimensional real hyperbolic space, 1⩽d⩽n−1, in terms of polynomials of the Laplace–Beltrami operator and intertwining fractional integrals. Similar results are established for hyperbolic cosine and sine transforms.