Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2??, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct fundamental domains for hyperbolic cone-manifold structures, from the geometry of a representation. Central to these techniques are the Euler class of a representation, the group ${\widetilde{PSL_{2}\mathbb{R}}}$ , the twist of hyperbolic isometries, and character varieties. We consider the action of the outer automorphism and related groups on the character variety, which is measure-preserving with respect to a natural measure derived from its symplectic structure, and ergodic in certain regions. Under various hypotheses, we almost surely or surely obtain a hyperbolic cone-manifold structure with prescribed holonomy.     

The geometry of the Gauss–Picard modular group

We give a construction of a fundamental domain for PU(2,1,\mathbbZ [i]){{\rm PU}(2,1,\mathbb{Z} [i])}, that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers \mathbbZ [i]{\mathbb{Z} [i]}. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.     

Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes

Let H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} denote the two dimensional hyperbolic space over \mathbb F{\mathbb F} , where \mathbb F{\mathbb F} is either the complex numbers \mathbb C{\mathbb C} or the quaternions \mathbb H{\mathbb H} . It is of interest to characterize algebraically the dynamical types of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} . For \mathbb F=\mathbb C{\mathbb F=\mathbb C} , such a characterization is known from the work of Giraud–Goldman. In this paper, we offer an algebraic characterization of isometries of H²_{\mathbb H}{{\bf H}^{\bf 2}_{\mathbb H}} . Our result restricts to the case \mathbb F=\mathbb C{\mathbb F=\mathbb C} and provides another characterization of the isometries of H²_{\mathbb C}{{\bf H}^{\bf 2}_{\mathbb C}} , which is different from the characterization due to Giraud–Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} and determine the z-classes.     

Algebraic characterization of the isometries of the hyperbolic 5-space

Let \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} be the group of invertible 2 × 2 matrices over the division algebra \mathbbH{\mathbb{H}} of quaternions. \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} .     

Codazzi spinors and globally hyperbolic manifolds with special holonomy

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface for any weakly irreducible holonomy representation with parallel spinors, t.m. with a holonomy group , where is trivial or a product of groups SU(k), Sp(l), G ₂ or Spin (7).      

Representation Formulas of Curves in a Two- and Three-Dimensional Lightlike Cone

In this paper we give the representation formulas for spacelike curves in two-dimensional lightlike cone ${{\mathbb Q}^2}$ and three-dimensional lightlike cone ${{\mathbb Q}^3}$ . Using these formulas we discuss the properties and structures of cone curves in ${{\mathbb Q}^2}$ and ${{\mathbb Q}^3}$ . Some examples are also given.     

Homogeneous p-adic vector bundles on abelian varieties that are analytic tori II

We investigate the temperate p-adic Riemann–Hilbert functor defined by André on abelian varieties that are analytic tori. We show that this functor induces an equivalence between the category of discrete and integral representation of the temperate fundamental group of the torus on finite dimensional \mathbb C_p{{\mathbb C}_{p}} -vector spaces as well as the category of homogeneous p-adic vector bundles on the torus.     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric g _h on given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (). By following the ideas and techniques by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group , arXiv:math.DG/0512547) we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot–Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C ² compact, connected, embedded surfaces in () with empty singular set and constant mean curvature H such that is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (). A. Hurtado has been partially supported by MCyT-Feder research project MTM2004-06015-C02-01. C. Rosales has been supported by MCyT-Feder research project MTM2004-01387.     

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy F^{^}{\Phi^\perp} (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \mathbbC Pⁿ{\mathbb{C} P^n}. Namely, if F^{^}{\Phi^\perp} does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \mathbbCⁿ{\mathbb{C}^n} the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \mathbbC Pⁿ{\mathbb{C} P^n} case) and basic facts of complex submanifolds.     

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H¹{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H¹{{\mathbb {H}}^1} the quantity \fracs_H(S?B(p,r))r^Q-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H¹{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.     

Minimal Lagrangian surfaces in {\mathbb {CH}^2}and representations of surface groups into SU(2, 1)

We use an elliptic differential equation of ?i?eica (or Toda) type to construct a minimal Lagrangian surface in ${\mathbb {CH}^2}$ from the data of a compact hyperbolic Riemann surface and a cubic holomorphic differential. The minimal Lagrangian surface is equivariant for an SU(2, 1) representation of the fundamental group. We use this data to construct a diffeomorphism between a neighbourhood of the zero section in a holomorphic vector bundle over Teichmuller space (whose fibres parameterise cubic holomorphic differentials) and a neighborhood of the ${\mathbb {R}}$ -Fuchsian representations in the SU(2, 1) representation space. We show that all the representations in this neighbourhood are complex-hyperbolic quasi-Fuchsian by constructing for each a fundamental domain using an SU(2, 1) frame for the minimal Lagrangian immersion: the Maurer–Cartan equation for this frame is the ?i?eica-type equation. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.     

Packings by translation balls in {\widetilde{{\rm SL}_2({\mathbb{R}})}}

For one of Thurston model spaces, \({\widetilde{{\rm SL}_2({\mathbb{R}})}}\) , we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particular, we found packings whose densities are close to the upper bound density for ball packings in the hyperbolic 3-space.     

Metrics with nonnegative curvature on {S^2 \times \mathbb{R}^4}

We study nonnegatively curved metrics on S²×\mathbbR⁴{S^2\times\mathbb{R}^4}. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking’s almost-positively curved metric on S ² × S ³ extends to a nonnegatively curved metric on S²×\mathbbR⁴{S^2\times\mathbb{R}^4} (so that Wilking’s space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.     

Sous-groupes discrets de <Emphasis Type="Italic">PU</Emphasis>(2,1) engendrés par <Emphasis Type="Italic">n</Emphasis> réflexions complexes et Déformation

Let PU(2,1) be the group of holomorphic isometries in the hyperbolic complex plane and let G _n be a sub-group of PU(2,1) which is generated by n complex reflections with respect to complex lines in . Under certain conditions, we prove that G _n is discrete. We construct representations ρ of the fundamental group Γ _g of the compact surface Σ _g of genus g, into PU(2,1), we prove they are discrete, faithful and we compute the dimension their deformation space.      

On the Isometry Groups of Invariant Lorentzian Metrics on the Heisenberg Group

This work concerns the invariant Lorentzian metrics on the Heisenberg Lie group of dimension three ${{\rm H}_3(\mathbb{R})}$ and the bi-invariant metrics on the solvable Lie groups of dimension four. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on ${{\rm H}_3(\mathbb{R})}$ by isometries and we study some geometrical features on these spaces. On ${{\rm H}_3(\mathbb{R})}$ , we prove that the property of the metric being proper naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons.     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S ¹ with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the corresponding contact structure. In particular, we prove that \mathbb CP²×S¹{{\mathbb {CP}}^2\times S^1} admits a contact structure in every homotopy class of almost contact structures. The existence of contact structures is also established for a large class of 5-manifolds with fundamental group \mathbbZ₂{{\mathbb{Z}}_2} .     

Foliations of some 3-manifolds which fiber over the circle

We show that a hyperbolic punctured torus bundle admits a foliation by lines which is covered by a product foliation. Thus its fundamental group acts freely on the plane.