Teichmuller Space Research Articles

Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure

Given a hyperbolic surface with geodesic boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmuller space T (S). We express the Weil-Petersson Poisson structure of T (S) in this system of coordinates and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.

Extension of the Weil-Petersson connection

Convexity properties of Weil-Petersson (WP) geodesics on the Teichmuller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson–Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions, the distance between horocycles for a hyperbolic metric

Harmonic mappings and moduli spaces of Riemann surfaces

Harmonic maps constitute a powerful tool for the analysis of moduli and Teichmuller spaces of compact Riemann surfaces. These moduli and Teichmuller spaces parametrize the different conformal structures a given compact differentiable surface F of some genus p can carry, together with a topological marking in the Teichmuller case. That is, each point in the moduli space Mp or the Teichmuller space Tp corresponds to some (marked) Riemann surface of genus p, that is, to some conformal structure on F . Mp then is a quotient of Tp by the mapping class group Γp, the group of homotopy classes of oriented diffeomorphisms of F . Since some nontrivial elements of Γp have fixed points, Mp acquires some quotient singularities. However, a suitable finite covering of Mp is free from singularities, that is, a manifold. Therefore, for many aspects, the singularities of Mp can be ignored, and this will sometimes simplify our discussion. – In this review, we shall confine ourselves mostly to the case p ≥ 2 which is the most difficult and most interesting case. Harmonic maps come from Riemannian geometry. They are defined as maps h : M → N between Riemannian manifolds that minimize a certain variational integral, called the energy. Thus, they depend on the Riemannian metrics of M and N . The theory works best when the metric of N has nonpositive sectional curvature. Harmonic maps can then be applied to moduli or Teichmuller spaces in two different ways. On one hand, one can look at harmonic maps to, from, or between Riemann surfaces and study how the harmonic maps or quantities associated to them, like their energy, depend on the underlying conformal structures. This is facilitated by the fact that the harmonic maps in question are unique in their homotopy classes, as we are looking at the case of genus p ≥ 2 which implies that the Riemann surfaces can be equipped with a hyperbolic metric, that is, one with constant negative curvature. In that manner,

Geometry of Teichmüller space with the Teichmüller metric

Geometry of Teichmüller space with the Teichmüller metric

Computing Fenchel-Nielsen Coordinates in Teichmüller Shape Space

Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to deform the surface conformally using surface Ricci flow, such that the Gaussian curvature is −1 everywhere. The surface is decomposed to several pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the practical value of the method for geometric database indexing, shape comparison and classification.

Cluster ensembles, quantization and the dilogarithm

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.

Quantization of the universal Teichmüller space

In the first part of this article we describe the complex geometry of the universal Teichmuller space, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The universal Teichmuller space contains classical Teichmuller spaces T(G), where G is a Fuchsian group, as complex submanifolds. The quotient Diff+(S 1)/Mob(S 1) of the diffeomorphism group of the unit circle modulo Mobius transformations can be considered as a ‘smooth’ part of the universal Teichmuller space. The second part is devoted to the quantization of the universal Teichmuller space. The smooth part Diff+(S 1)/Mob(S 1) may be quantized, using its embedding into the Hilbert–Schmidt Grassmannian. However, this quantization method does not apply to the whole universal Teichmuller space, to which the ‘quantized calculus’ of Connes and Sullivan may be applied.

Inequalities on the inner radius of univalency and the norm of pre-Schwarzian derivative

In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.

Image of asymptotic Bers map

In this paper, we give a geometric characterization for developing mappings such that the asymptotic class of its Schwarzian derivative is in the image of the asymptotic Bers map from the asymptotic Teichmuller space of the unit disk D . We also give a characterization of points in the closure of the image, and discuss the density problem for the asymptotic Teichmuller space.

Electrical Impedance Tomography Problem With Inaccurately Known Boundary and Contact Impedances

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmUller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space

The aim of this paper is to develop the theory of a compactification of Teichmuller space given by F. Gardiner and H. Masur, which we call the Gardiner–Masur compactification of the Teichmuller space. We first develop the general theory of the Gardiner–Masur compactification. Secondly, we will investigate the asymptotic behaviors of Teichmuller geodesic rays under the Gardiner–Masur embedding. In particular, we will observe that the projective class of a rational measured foliation G can not be an accumulation point of every Teichmuller geodesic ray under the Gardiner–Masur embedding, when the support of G consists of at least two simple closed curves.

The Weil–Petersson geometry of the moduli space of Riemann surfaces

In 2007, Z. Huang showed that in the thick part of the moduli space M g of compact Riemann surfaces of genus g, the sectional curvature of the Weil-Petersson metric is bounded below by a constant depending on the injectivity radius, but independent of the genus g. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichmuller space equipped with a Hilbert structure induced by the Weil-Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.

The quantum dilogarithm and representations of quantum cluster varieties

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

Lines of minima are uniformly quasigeodesic

We continue the comparison between lines of minima and Teichmuller geodesics begun in our previous work. For two measured laminations nu(+) and nu(-) that fill up a hyperbolizable surface S and for t is an element of (-infinity, infinity), let L-t be the unique hyperbolic surface that minimizes the length function e(t) l(nu(+)) + e(-t) l(nu(-)) on Teichmuller space. We prove that the path t bar right arrow L-t is a Teichmuller quasigeodesic.

On the equivalence of extremal Teichmüller mapping

Let T(Δ) and B(Δ) be the Teichmuller space and the infinitesimal Teichmuller space of the unit disk Δ respectively. In this paper, we show that [ν]B(Δ) being an infinitesimal Strebel point does not imply that [ν]T(Δ) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved.

Lines of Minima and Teichmüller Geodesics

For two measured laminations ν+ and ν− that fill up a hyperbolizable surface S and for \(t\,\in\,(-\infty,\infty)\), let \({\mathcal{L}}_t\) be the unique hyperbolic surface that minimizes the length function etl(ν+) + e-tl(ν−) on Teichmuller space. We characterize the curves that are short in \({\mathcal{L}}_t\) and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface \({\mathcal{G}}_t\) on the Teichmuller geodesic whose horizontal and vertical foliations are respectively, etν+ and etν−. By deriving additional information about the twists of ν+ and ν− around the short curves, we estimate the Teichmuller distance between \({\mathcal{L}}_t\) and \({\mathcal{G}}_t\). We deduce that this distance can be arbitrarily large, but that if S is a once-punctured torus or four-times-punctured sphere, the distance is bounded independently of t.

Moyal-Weyl star product and quasiconformal mapping relationship

The deformation quantization procedure can introduce the quantum effect by including the deformation of complex structures on a bi-dimensional Riemann surface Σ without boundary. This is the platform of bi-dimensional conformal models like string theory and two-dimensional gravity. Then, the star product can become a quasiconformal mapping on this surface. This makes the road toward a Teichmuller theory of the Moyal-Weyl star product through the quasiconformal mappings as lifting of universal covering surfaces of any Riemann surface.

Behavior of geodesic-length functions on Teichmüller space

Let T be the Teichmuller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichmuller space of marked Riemann surfaces with nodes, (Abi77, Ber74). Pro- vided with the WP metric T is a complete CAT(0) metric space, (DW03, Wol03, Yam04). An invariant of a marked hyperbolic struc- ture is the lengthof the geodesic in a free homotopy class. A basic feature of Teichmuller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop the understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include ex- pansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bound- ing configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a dis- crete group of isometries and bounding sums of the inverse square exponential-distance.

Dimension and rank for mapping class groups

We study the large scale geometry of the mapping class group, MCG(S). Our main result is that for any asymptotic cone of MCG(S), the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S). An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG(S) has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric.

The limiting shape of one-dimensional Teichmüller spaces

We show that the Bers embedding of the Teichmuller space of a once-punctured torus converges to the cardioid in the sense of Caratheodory up to rotation when the base torus goes to the boundary of its moduli space.