Real Hyperbolic Manifolds Research Articles

Rigidity of flat holonomies

Abstract We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension $n \geq 3$ with strongly $1/4$ -pinched or relatively $1/2$ -pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds

In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality.

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.

Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold

In this work, we study the existence and stability of asymptotically almost periodic mild solutions of vectorial parabolic equations on the real hyperbolic manifold Hd(R) (d⩾2). We will consider the equations associated with Ebin-Marsden's Laplace operator. Our method is based on certain dispersive and smoothing estimates of the semigroup associated with the linearized vectorial heat equation and the fixed point argument. First, we prove the existence and uniqueness of the asymptotically almost periodic mild solution for the linear equation. Next, using the fixed point argument we can pass from the linear equation to the semi-linear equation to prove the existence and uniqueness of such solution. We prove also the exponential stability of solution by using the cone inequality. Finally, abstract results will be applied to Navier-Stokes equations and the semi-linear vectorial heat equation.

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

AbstractWe prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

Boundary rigidity of convex cocompact real hyperbolic manifolds

The goal of this short paper is to show that the hyperbolic metric is uniquely determined by the induced metric on the strictly convex smooth boundary for dimension ≥ 4 \geq 4 .

A couple of real hyperbolic disc bundles over surfaces

Applying the techniques developed in [1], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov–Lawson–Thurston conjecture [6], such bundles M \to S should satisfy the inequality |eM/\chi S|\leq 1 , where eM stands for the Euler number of the bundle and \chi S , for the Euler characteristic of the surface. In this paper, we construct new examples that provide a maximal value of |eM/\chi S|=\frac {3}{5} among all known examples. The former maximum, belonging to Feng Luo [10], was |eM/\chi S|=\frac {1}{2} .

Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρnlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

Gallagherian Prime Geodesic Theorem in Higher Dimensions

Using the Gallagher–Koyama approach, we reduce the exponent in the error term of the prime geodesic theorem for real hyperbolic manifolds with cusps.

Convex real projective structures and Weil's local rigidity theorem

For an n-dimensional real hyperbolic manifold M, we calculate the Zariski tangent space of a character variety χ(π1(M),SL(n+1,R)), n>2 at Fuchsian locus to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniform hyperbolic lattices using real projective structures.

Upper bounds for geodesic periods over rank one locally symmetric spaces

Abstract We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

Closed geodesics and holonomies for Kleinian manifolds

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup \({\Gamma}\) of G, we establish the joint equidistribution of closed geodesics and their holonomy classes for the associated locally symmetric space. Our result is given in a quantitative form for geometrically finite real hyperbolic manifolds whose critical exponents are big enough. In the case when \({G={\rm PSL}_2 (\mathbb{C})}\) , our results imply the equidistribution of eigenvalues of elements of Γ in the complex plane.

Hyperfunctions in hyperbolic geometry

In the framework of real hyperbolic geometry, this review note begins with the Helgason correspondence induced by the Poisson transform between eigenfunctions of the Laplace–Beltrami operator on the hyperbolic space and hyperfunctions on its boundary at infinity . Focused on the scattering operator for real hyperbolic manifolds of finite geometry, discussion is given on the two different constructions (pseudo-differential calculus for degenerate operators and harmonic analysis for the conformal group) and some applications (Selberg zeta functions, resonances and scattering poles).

Rigidity and L^2 cohomology of hyperbolic manifolds

When X=Γ∖ℍ n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps

In this paper, we study the Ruelle zeta function and the Selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. In particular, we show that they have meromorphic extensions to \({\mathbb{C}}\) and satisfy functional equations. We also derive the order of the singularity of the Ruelle zeta function at the origin. To prove these results, we completely analyze the weighted unipotent orbital integrals on the geometric side of the Selberg trace formula when test functions are defined for the fundamental representations.

Hyperbolic manifolds, amalgamated products and critical exponents

We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of dim n is a free product of its subgroups A and B over the amalgamated subgroup C, then the critical exponent of C is not smaller than n−2. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

The double of a hyperbolic manifold and non-positively curved exotic $PL$ structures

We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable PL structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.

DETERMINANT LINE BUNDLES AND TOPOLOGICAL INVARIANTS OF HYPERBOLIC GEOMETRY

Some remarks on determinant line bundles and topological invariants are given. The index of the twisted Dirac operator acting on real hyperbolic manifolds is computed. We briefly discuss physical applications of results.

Equidistribution of holonomy restricted to a homology class about closed geodesics

The purpose of this paper is, for each homology class, to prove the equidistribution theorem of restricted holonomy classes about closed geodesics mainly for the real hyperbolic manifolds. More precisely, this work is a continuation of the study [SW] (Duke Math. J. 100 (1999), 1–57) but one restricts these holonomy classes to the ones arising from the subset of closed geodesics whose corresponding hyperbolic conjugacy classes belong to a given homology class of the manifold. As an application, we also discuss the equidistribution of (imaginary) quadratic forms with respect to the argument of fundamental units. 1991 Mathematical Subject Classification. Primary 22; Secondary 53.

Bending deformations of complex hyperbolic surfaces.

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, our main goal is to show that they enjoy nevertheless the flexibility of low-dimensional real hyperbolic manifolds. Namely we define a class of ``bending deformations of a given (Stein) complex surface $M$ associated with its closed geodesics provided that $M$ is homotopy equivalent to a Riemann surface whose embedding in $M$ has a non-trivial totally real geodesic part. Such bending deformations bend $M$ along its closed geodesics and are induced by equivariant quasiconformal homeomorphisms of the complex hyperbolic space and its Cauchy-Riemannian structure at infinity.