Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

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 2π, 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 prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group [(PSL₂\mathbb R)\tilde]{\widetilde{{\it PSL}_2{\mathbb R}}} of the group of orientation-preserving isometries of \mathbb H²{{\mathbb H}^2} and Markoff moves arising from the action of the mapping class group on the character variety.     

On the full holonomy group of Lorentzian manifolds

The classification of restricted holonomy groups of \(n\) -dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.     

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.     

Surfaces in {\mathbb{R}^4} with constant principal angles with respect to a plane

We study surfaces in ${\mathbb{R}^4}$ whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of ${\mathbb{R}^2}$ . We classify all surfaces with one principal angle equal to 0 and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in ${\mathbb{R}^4}$ with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector field.     

Surfaces of M_k^2\times \mathbb R invariant under a one-parameter group of isometries

We assume that an immersed constant mean curvature surface ${\varSigma } \looparrowright {M_k} \times \mathbb R $ satisfies a relation involving the mean curvature, the Gaussian curvature and the angle that the unit vector of the factor $\mathbb R $ makes with the normal to the surface. This relation, although given initially in its pointwise form, can be shown to be equivalent to an integral relation. From the assumed relation, it follows that $\varSigma $ is invariant under a one-parameter group of isometries of $M_k^2\times \mathbb{R }$ which are induced by the isometries of $M_k^2$  . An application is made to describe qualitatively those surfaces for which the Abresch-Rosenberg complex quadratic form vanishes.     

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc $\mathcal {D}$ whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in $\mathcal {D}$ , which is invariant under the action of a lattice subgroup ?? of U(1,1), the group of isometries of ${\mathcal{D}}$ . In our case ?? generates a tiling of $\mathcal {D}$ with regular octagons. This problem was introduced as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under ??generic?? assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms $\mathcal {G}$ of the compact Riemann surface $\mathcal {D}/\varGamma $ . The irreducible representations of $\mathcal {G}$ have dimensions one, two, three and four. The bifurcation diagrams for the representations of dimensions less than four have already been described and correspond to well-known group actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.     

On volumes of hyperbolic Coxeter polytopes and quadratic forms

In this paper, we compute the covolume of the group of units of the quadratic form ${f_d^n(x) = x_1^2+x_2^2+ \cdots +x_n^2-dx_{n+1}^2}$ with d an odd, square-free, positive integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form ${f_3^n}$ . We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.     

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and second variation formulae for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that move the singular set of a ${\mathcal{C}^2}$ surface and non-singular variations for ${\mathcal{C}^2_\mathcal{H}}$ surfaces. These formulae enable us to construct a stability operator for non-singular ${\mathcal{C}^2}$ surfaces and another one for ${\mathcal{C}^2}$ (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in terms of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we give a classification of the complete stable surfaces in the roto-translation group ${\mathcal{R}\mathcal{T}}$ .     

Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

Simply connected three-dimensional homogeneous manifolds ${\mathbb{E}(\kappa, \tau)}$ , with four-dimensional isometry group, have a canonical Spin^c structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into ${\mathbb{E}(\kappa, \tau)}$ . As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in ${\mathbb{E}(\kappa, \tau)}$ . Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin^c spinors.     

Equivariance and extendibility in finite reductive groups with connected center

We show that several character correspondences for finite reductive groups $G$ are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to $G$ has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broué–Malle–Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs–Malle–Navarro for the non-abelian finite simple groups of Lie types $^3\mathsf{D }_4,\mathsf{E }_8,\mathsf{F }_4,^2\mathsf{F }_4$ , and $\mathsf{G }_2$ .     

On the cohomology of moduli spaces of (weighted) stable rational curves

We give a recursive algorithm for computing the character of the cohomology of the moduli space ${\overline{M}}_{0,n}$ of stable $n$ -pointed genus zero curves as a representation of the symmetric group $\mathbb{S }_n$ on $n$ letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.     

The Lagrangian Radon Transform and the Weil Representation

We consider the operator $\mathcal {R}$ , which sends a function on ${\mathbb {R}}^{2n}$ to its integrals over all affine Lagrangian subspaces in ${\mathbb {R}}^{2n}$ . We discuss properties of the operator $\mathcal {R}$ and of the representation of the affine symplectic group in several function spaces on ${\mathbb {R}}^{2n}$ .     

Fuchsian convex bodies: basics of Brunn–Minkowski theory

The hyperbolic space ${\mathbb{H}^d}$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that ${\mathbb{H}^d/\Gamma}$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.     

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space $ \mathcal{M} $ of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, $ H_\mathbb{C}^n $ , up to its holomorphic isometry group PU(n, 1): One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $ \mathcal{M} $ . For n = 2, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $ \mathcal{M} $ for any dimension n ≥ 1.     

Crystallographic number systems

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion $$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$ for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.     

Cabling procedure for the colored HOMFLY polynomials

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$ -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$ -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$ -matrices and clarifying some conjectures formulated in previous papers.     

Split Partial Isometries

A partial isometry V is said to be a split partial isometry if ${\mathcal{H}=R(V) + N(V)}$ , with R(V) ∩ N(V) = {0} (R(V) = range of V, N(V) = null-space of V). We study the topological properties of the set ${\mathcal{I}_0}$ of such partial isometries. Denote by ${\mathcal{I}}$ the set of all partial isometries of ${\mathcal{B}(\mathcal{H})}$ , and by ${\mathcal{I}_N}$ the set of normal partial isometries. Then $$\mathcal{I}_N\subset \mathcal{I}_0\subset \mathcal{I}, $$ and the inclusions are proper. It is known that ${\mathcal{I}}$ is a C ^∞-submanifold of ${\mathcal{B}(\mathcal{H})}$ . It is shown here that ${\mathcal{I}_0}$ is open in ${\mathcal{I}}$ , therefore is has also C ^∞-local structure. We characterize the set ${\mathcal{I}_0}$ , in terms of metric properties, existence of special pseudo-inverses, and a property of the spectrum and the resolvent of V. The connected components of ${\mathcal{I}_0}$ are characterized: ${V_0,V_1\in \mathcal{I}_0}$ lie in the same connected component if and only if $${\rm dim}\, R(V_0)= {\rm dim}\, R(V_1) \,\,{\rm and}\,\,\, {\rm dim}\, R(V_0)^\perp = {\rm dim}\, R(V_1)^\perp.$$ This result is known for normal partial isometries.     

{SL(2, \mathbb{R})} -invariant probability measures on the moduli spaces of translation surfaces are regular

In the moduli space ${{\mathcal {H}}_g}$ of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ? ρ. We prove that this subset of ${{\mathcal {H}}_g}$ has measure o(ρ ²) w.r.t. any probability measure on ${{\mathcal {H}}_g}$ which is invariant under the natural action of ${SL(2,\mathbb{R})}$ . This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin–Kontsevich–Zorich on the Lyapunov exponents of the KZ-cocycle.     

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. When \({\Gamma}\) is a lattice, the equidistribution of holonomies was proved by Sarnak and Wakayama in 1999 using the Selberg trace formula.