Spherical CR Structures Research Articles

Chern classes of spherical CR manifolds

We first construct closed spherical CR manifolds of dimension at least five having non-trivial first Chern class with real coefficients. We next prove a constraint on Chern classes with real coefficients of (not necessarily closed) spherical CR manifolds. Finally, we obtain a topological obstruction to the existence of spherical CR structures on co-oriented contact manifolds.

Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane $\mathbb{H}^2_{\mathbb{C}}$. We deform the Ford domain of Parker and Will in $\mathbb{H}^2_{\mathbb{C}}$ in a one parameter family. On the one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer $n \geq 4$, and the spherical CR structure obtained for $n = 4$ is the Deraux-Falbel spherical CR uniformization of the Figure Eight knot complement.

Connected Sum of Spherical CR Manifolds with Positive CR Yamabe Constant

Suppose \(M_{1}\) and \(M_{2}\) are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of \(M_{1}\) and \(M_{2}\) also admits a spherical CR structure with positive CR Yamabe constant.

Complex hyperbolic geometry of the figure-eight knot

We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.

Branched sperical CR structures on the complement of the figure-eight knot

Abstract. We obtain a branched spherical CR structure on the com- plement of the figure-eight knot whose holonomy representation was given in [4]. There are essentially two boundary unipotent representations from the complement of the figure-eight knot into PU(2,1), which we call ρ1 and ρ2. We make explicit some fundamental differences between these two representations. For instance, seeing the figure-eight knot complement as a surface bundle over the circle, the behavior of the fundamental group of the fiber under the representa- tion is a key difference between ρ1 and ρ2.

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kahler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Englis (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.

A spherical CR structure on the complement of the figure eight knot with discrete holonomy

We describe a general geometrical construction of representations of fundamental groups of 3-manifolds into PU(2, 1) and eventually of spherical CR structures defined on those 3-manifolds. We construct branched spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2, 1,Z[ω]) and PU(2, 1,Z[i]) respectively.

Geometric structures associated to triangulations as fixed point sets of involutions

We establish that hyperbolic structures and spherical CR structures on a three-dimensional manifold are contained in fixed point sets of a larger class of structures associated to a triangulation of the manifold. We generalize the 5 term relation to this setting.

The Teichmüller distance on the space of spherical CR structures

We define the Teichmuller pseudodistance on the space of spherical CR structures on a fixed compact manifold by using quasiconformal mappings between spherical CR manifolds. The pseudodistance is shown to be a complete distance.

Constructing spherical CR manifolds by gluing tetrahedra

We propose a general method of constructing spherical CR manifolds by gluing tetrahedra adapted to CR geometry. We obtain spherical CR structures on the complement of the figure eight knot and the Whitehead link complement with holonomy in PU ( 2 , 1 , Z [ ω ] ) and PU ( 2 , 1 , Z [ i ] ) respectively (the same integer rings appearing in real hyperbolic geometry). To cite this article: E. Falbel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

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.

Flexibility of ideal triangle groups in complex hyperbolic geometry

We show that the Teichmüller space of the ideal triangle group in the automorphism group of complex hyperbolic space contains a real four-dimensional ball. This implies the existence of a four-dimensional family of spherical CR structures on the trivial circle bundle over the sphere minus three points. The proof is an explicit construction of fundamental domains whose boundaries are special hypersurfaces foliated by complex geodesics.

Deformation of spherical CR structures and the universal Picard variety

We study deformation of spherical $CR$ circle bundles over Riemann surfaces of genus > 1. There is a one to one correspondence between such deformation space and the so-called universal Picard variety. Our differential-geometric proof of the structure and dimension of the unramified universal Picard variety has its own interest, and our theory has its counterpart in the Teichmuller theory.

Rigidity of automorphisms and spherical CR structures

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function Λ onM. AsM supports a contact form, there exists a characteristic vector field ξ dual to the contact structure. If ξ induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature Λ or of nonpositive curvature Λ. By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature Λ turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature Λ (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When Λ≤0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.

A combinatorial invariant for spherical CR structures

We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal{P}(\mathbb{C})$. If $M$ is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal{P}(\mathbb{C})$-valued invariant for $M$. We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on $M$, we show the invariant lies in the Bloch group $\mathcal{B}(k)$, where $k$ is the field generated by the cross-ratio. For a CR triangulation of the Whitehead link complement, we show its invariant is a torsion in $\mathcal{B}(k)$ and for a triangulation of the complement of the 52-knot we show that the invariant is not trivial and not a torsion element.