Kleinian Manifolds Research Articles

Nonlocal curvature and topology of locally conformally flat manifolds

In this paper, we focus on the geometry of compact conformally flat manifolds (Mn,g) with positive scalar curvature. Schoen–Yau proved that its universal cover (Mn˜,g˜) is conformally embedded in Sn such that Mn is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension <n−22. If additionally we assume that the non-local curvature Q2γ≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to n−2γ2. If Q2γ>0, then the above inequality is strict. Moreover, the above upper bound is sharp. As applications, we obtain some topological rigidity and classification theorems in lower dimensions.

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.

Constructing geometrically infinite groups on boundaries of deformation spaces

Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold M = ( H 3 ⋃ Ω G ) / G . Suppose that for each boundary component of M , either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit Γ of quasi-conformal deformations of G such that there is a homeomorphism h from Int M to H 3 / Γ compatible with the natural isomorphism from G to Γ , the given laminations are unrealisable in H 3 / Γ , and the given conformal structures are pushed forward by h to those of H 3 / Γ . Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden’s conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.

Invariance of the Nayatani metrics for Kleinian manifolds

The Nayatani metric g N is a Riemannian metric on a Kleinian manifold M which is compatible with the standard flat conformal structure. It is known that, for M corresponding to a geometrically finite Kleinian group, g N has large symmetry: the isometry group of (M, g N ) coincides with the conformal transformation group of M. In this paper, we prove that this holds for a larger class of M. In particular, this class contains such M that correspond to Kleinian groups of divergence type.

Nayatani’s metric and conformal transformations of a Kleinian manifold

According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold M M which is compatible with the canonical flat conformal structure. He showed that this metric g N g_N has a large symmetry if g N g_N is a complete metric. Under certain assumptions including the completeness of g N g_N , the isometry group of ( M , g N ) (M,g_N) coincides with the conformal transformation group of M M . In this paper, we show that g N g_N may have a large symmetry even if g N g_N is not complete. In particular, every conformal transformation is an isometry when ( M , g N ) (M,g_N) corresponds to a geometrically finite Kleinian group.

On finiteness of Kleinian groups in general dimension

Abstract. In this paper we provide a criteria for geometric finiteness of Kleiniangroups in general dimension. We formulate the concept of conformal finiteness forKleinian groups in space of dimension higher than two, which generalizes the notionof analytic finiteness in dimension two. Then we extend the argument in the paperof Bishop and Jones to show that conformal finiteness implies geometric finitenessunless the set of limit points is of Hausdorff dimension n. Furthermore we show that,for a given Kleinian group Γ, conformal finiteness is equivalent to the existence of ametric of finite geometry on the Kleinian manifold Ω(Γ)/Γ. 1991 Mathematics Subject Classification: Primary 53A30; Secondary 30F40,58J60, 53C21.§0. IntroductionA discrete subgroup Γ of the group of conformal transformations of the unitsphere S n is called a Kleinian group if there is a non-empty domain Ω(Γ) of dis-continuity in S n . The group Γ also acts as a subgroup of the group of hyperbolicisometries of the unit ball B n+1

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold OHgr(Gamma)/Gamma, where Gamma is a convex cocompact subgroup of AutCRS2n+1=PU(n+1,1) and OHgr(Gamma) is the discontinuity domain of Gamma. This contact form can be used to prove that OHgr(Gamma)/Gamma is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent delta(Gamma)n, or delta(Gamma)=n). This generalizes Nayatanirsquos result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.

The Spectrum of Kleinian Manifolds

We obtain the Plancherel theorem for L2(Γ\\G), where G is a classical simple Lie group of real rank one and Γ⊂G is convex–cocompact discrete subgroup, and deduce its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. As the main tool, we develop a geometric version of scattering theory which, in particular, contains the meromorphic continuation of the Eisenstein series for this situation. The central role played by invariant distribution sections supported on the limit set is emphasized.

Obstructions to pin structures on Kleinian manifolds

We develop various topological notions on four-manifolds of Kleinian signature (−−++). In particular, we extend the concept of ‘‘Kleinian metric homotopy’’ to nonorientable manifolds. We then derive the topological obstructions to pin-Klein cobordism for all of the pin groups. Finally, we discuss various examples and applications which arise from this work.

Spin structures on Kleinian manifolds

We derive the topological obstruction to spin--Klein cobordism. This result has implications for signature change in general relativity, and for the N=2 superstring.