Teichmuller Space Research Articles

Basic $$\mathop {\mathrm {PU}}(1,1)$$-representations of the hyperelliptic group are discrete

We show that a \(\mathop {\mathrm {PU}}(1,1)\)-representation of the hyperelliptic group \(H_n\) is basic if and only if it is discrete and faithful, thus partially proving a conjecture by S. Anan’in (A hyperelliptic view on Teichmuller space. II, 2009. arXiv:0709.1711) and S. Anan’in and E. Bento Gonçalves (A hyperelliptic view on Teichmuller space. I, 2007. arXiv:0907.1633) in the case of the Poincaré disc.

On the Carathéodory metric of universal Teichmüller space

In contrast to finite dimensional Teichmuller spaces, all non-expanding invariant metrics on the universal Teichmuller space coincide. This important fact found various applications. We give its new, simplified proof based on some deep features of the Grunsky operator, which intrinsically relate to the universal Teichmuller space. This approach also yields a quantitative answer to Ahlfors' question.

Plurisubharmonicity and geodesic convexity of energy function on Teichmueller space

Plurisubharmonicity and geodesic convexity of energy function on Teichmueller space

Generalizations of the Eierlegende–Wollmilchsau

We classify a natural collection of GL(2,R)-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmuller space. This classification is the key input in subsequent work of the authors that classifies high rank invariant subvarieties, and in subsequent work of the first author that classifies invariant subvarieties with Lyapunov spectrum as degenerate as possible and not consisting of torus covers. We also derive applications to the complex geometry of Teichmuller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.

Presentations of the Roger–Yang generalized skein algebra

We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.

Teichmüller space theory and classical problems of geometric function theory

Recently the author presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmuller spaces. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions

Quantitative Characterization of the Human Retinotopic Map Based on Quasiconformal Mapping

Abstract The retinotopic map depicts the cortical neurons’ response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmuller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a “Beltrami coefficient map” (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

Teichmuller space theory and classical problems of geometric function theory

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions.

Cluster realizations of Weyl groups and higher Teichmüller theory

For a symmetrizable Kac–Moody Lie algebra $${\mathfrak {g}}$$ , we construct a family of weighted quivers $$Q_m({\mathfrak {g}})$$ ( $$m \ge 2$$ ) whose cluster modular group $$\Gamma _{Q_m({\mathfrak {g}})}$$ contains the Weyl group $$W({\mathfrak {g}})$$ as a subgroup. We compute explicit formulae for the corresponding cluster $${{\mathcal {A}} }$$ - and $${{\mathcal {X}} }$$ -transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for $$Q_m({\mathfrak {g}})$$ in a systematic way when $${\mathfrak {g}}$$ is of finite type. Moreover if $${\mathfrak {g}}$$ is of classical finite type with the Coxeter number h, the quiver $$Q_{kh}({\mathfrak {g}})$$ ( $$k \ge 1$$ ) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmuller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmuller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.

Optimal Lipschitz maps on one-holed tori and the Thurston metric theory of Teichmüller space

We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmuller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmuller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is $$\le 4{\text {arcsinh}}(1)$$ , but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmuller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

On Teichmüller space of circle diffeomorphisms with Hölder continuous derivative

Matsuzaki [M1] introduced the Teichmuller space $$T_{0}^{\alpha }$$ of diffeomorphisms of the unit circle with Holder continuous derivatives and investigated its Schwarzian derivative model. This paper deals with the pre-Schwarzian derivative model $$T_{0}^{\alpha }(1)$$ of the Teichmuller space $$T_{0}^{\alpha }$$ . It is shown that $$T_{0}^{\alpha }(1)$$ is a connected open subset of $${\mathcal {B}}_{0}^{\alpha }(\Delta )$$ and the pre-Bers projection is a holomorphic split submersion in $$T_{0}^{\alpha }$$ .

Universal commensurability augmented Teichmüller space and moduli space

It is known that every unbranched finite covering \(\alpha\colon\widetilde{S}_{g(\alpha)}\rightarrow S\) of a compact Riemann surface \(S\) with genus \(g\geq 2\) induces an isometric embedding \(\Gamma_{\alpha}\) from the Teichmuller space \(T(S)\) to the Teichmuller space \(T(\widetilde{S}_{g(\alpha)})\). Actually, it has been showed that the isometric embedding \(\Gamma_{\alpha}\) can be extended isometrically to the augmented Teichmuller space \(\widehat{T}(S)\) of \(T(S)\). Using this result, we construct a direct limit \(\widehat{T}_{\infty}(S)\) of augmented Teichmuller spaces, where the index runs over all unbranched finite coverings of \(S\). Then, we show that the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) can extend isometrically on \(\widehat{T}_{\infty}(S)\). Furthermore, for any \(X_{\infty}\in T_{\infty}(S)\), its orbit of the action of the universal commensurability modular group \(\operatorname{Mod}_{\infty}(S)\) on \(\widehat{T}_{\infty}(S)\) is dense. Finally, we also construct a direct limit \(\widehat{M}_{\infty}(S)\) of augmented moduli spaces by characteristic towers and show that the subgroup \(\operatorname{Caut}(\pi_{1}(S))\) of \(\operatorname{Mod}_{\infty}(S)\) acts on \(\widehat{T}_{\infty}(S)\) to produce \(\widehat{M}_{\infty}(S)\) as the quotient.

On continuity of drifts of the mapping class group

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the `translation distance' of the random walk. In this paper, we prove that the drift varies continuously with the transition probability measures, under the assumption that the distance and the horofunctions on $X$ are expressed by certain ratios. As an example, we consider the mapping class group $\mathrm{MCG}(S)$ acting on the Teichmuller space. By using north-south dynamics, we also consider the continuity of the drift for a sequence converging to a Dirac measure. As an appendix, we prove that the asymptotic entropy of the random walks on $\mathrm{MCG}(S)$ varies continuously.

Rigidity of Teichmüller space

We prove the holomorphic rigidity conjecture of Teichmuller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmuller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth n-dimensional Riemannian domain to the Weil–Petersson completion $$\overline{\mathcal {T}}$$ of Teichmuller space has Hausdorff dimension at most $$n-2$$ , and moreover, u has certain decay near the singular set. Combining this with the earlier work of Schumacher, Siu and Jost-Yau, we provide a proof of the holomorphic rigidity of Teichmuller space. In addition, our results provide as a byproduct a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by Farb–Masur and Yeung.

Riemannian metrics on the moduli space of GHMC anti-de Sitter structures

We first extend the construction of the pressure metric to the deformation space of globally hyperbolic maximal Cauchy-compact anti-de Sitter structures. We show that, in contrast with the case of the Hitchin components, the pressure metric is degenerate and we characterize its degenerate locus. We then introduce a nowhere degenerate Riemannian metric adapting the work of Qiongling Li on the $$\mathrm {SL}(3,\mathbb {R})$$ -Hitchin component to this moduli space. We prove that the Fuchsian locus is a totally geodesic copy of Teichmuller space endowed with a multiple of the Weil–Petersson metric.

Basmajian-type inequalities for maximal representations

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian’s identity from Teichmuller theory. Any equality characterizes diagonal embeddings.

Laurent phenomenon algebras arising from surfaces II: Laminated surfaces

It was shown by Fock and Goncharov (Dual Teichmuller and lamination spaces. Handbook of Teichmuller Theory, 2007), and Fomin et al. (Acta Math 201(1):83–146, 2008) that some cluster algebras arise from triangulated orientable surfaces. Subsequently, Dupont and Palesi (J Algebraic Combinatorics 42(2):429–472, 2015) generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. In Wilson (Int Math Res Notices 341, 2017) we linked this framework to Lam and Pylyavskyy’s Laurent phenomenon algebras (J Math 4(1):121–162, 2016), showing that unpunctured surfaces admit an LP structure. In this paper we extend quasi-cluster algebras to include punctured surfaces. Moreover, by adding laminations to the surface we demonstrate that all punctured and unpunctured surfaces admit LP structures. In short, we link two constructions which arose as seemingly unrelated generalisations of cluster algebras—one of the generalisations (quasi-cluster algebras) being based on triangulated surfaces, and the other (Laurent phenomenon algebras) based on the Laurent phenomenon. We thus provide a rich class of geometric examples in which to help study Laurent phenomenon algebras.

Teichmüller spaces and coefficient problems for univalent holomorphic functions

We show that the features of Teichmuller spaces give rise to a completely new powerful tool for solving the classical coefficient problems in geometric function theory. Here we concern the univalent functions. Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $$J(f) = J(a_{m_1}, a_{m_2},\dots , a_{m_n}) $$ on some classes of univalent functions in the unit disk naturally connected with the canonical class S. The given functional J is lifted to the Teichmuller space $$\mathbf{T}_1$$ of the punctured disk $$\mathbb {D}_{*} = \{0< |z| < 1\}$$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichmuller space. This generates a positive subharmonic function on the disk $$\{|t| < 4\}$$ with $$\sup _{|t|<4} u(t) = \max _{{\mathbf {T}}_1} |J|$$ attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function. This theorem implies new sharp distortion estimates for univalent functions giving explicitly the extremal functions, and creates a new bridge between Teichmuller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

In this paper, we extend the construction of pressure metrics to Teichmuller spaces of surfaces with punctures. This construction recovers Thurston’s Riemannian metric on Teichmuller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

On the Compactness of Grunsky Differential Operators

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmuller spaces. The Grunsky map is known to be holomorphic on the universal Teichmuller space. In this paper the authors deal with the compactness of a Grunsky differential operator. They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator.