# Questions tagged [teichmuller-theory]

The teichmuller-theory tag has no usage guidance.

234
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### Metric balls in Teichmüller space are topological balls

Let $X$ be a topological surface of finite type and $\mathcal{T}_X$ be the corresponding Teichmüller space. Let $B$ be a ball with respect to the Teichmüller metric on $\mathcal{T}_X$ (i.e., the ...

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### Teichmuller interpretation of unbounded holomorphic quadratic differentials

For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...

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### Teichmuller space and almost complex structures

Let $\Sigma$ be a closed orientable surface of genus $g$. It is well known that every almost complex structure on a surface is induced by a complex atlas. Therefore, if we call $\mathcal{J}(\Sigma)$ ...

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### What is known about the almost complex structure on the Teichmüller space in Fenchel–Nielsen coordinates?

There has been a question on the same subject, but I'm asking about something more specific.
In the Fenchel–Nielsen coordinates, the Teichmüller space of genus $g$ is represented as $\mathbb{R}^{3g-3}\...

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### Representation determined by traces

A discrete, faithful representation of a surface group $G:\pi_1(S_g) \to PSL_2(\mathbb{R})$ is determined, up to conjugacy by $PGL_2(\mathbb R)$, among such representations by the squares of traces of ...

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### Computing some closed trajectories of meromorphic quadratic differentials

I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; ...

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### Intersection of orbits of earthquake flow on Teichmüller space

Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...

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### Further directions in representations of surface group into a Lie group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...

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### Uniformization of Riemann surfaces with cone singularities

Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such ...

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### Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...

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### Is there an extension of Ogg's results to surfaces of Genus 1

The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...

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### Does moving a small enough distance in Teichmüller space change the marking?

Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \...

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### Has anyone studied the PDE generalization of Teichmüller Space?

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize).
Given a surface $S$ we can define Teichmüller space $T(S)$ to ...

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### Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map
\begin{align*}
\pi\...

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### Fenchel–Nielsen coordinates vs Fock–Goncharov coordinates

Consider an orientable surface $S$ and its Teichmüller space $S$, which is the space of representations of its fundamental group $T(S)=\{\rho: \pi_1(S) \to \operatorname{SL}(2,\mathbb{R})\}$. Fock and ...

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### Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...

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### The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...

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### Automorphism group of moduli space $M_{0,n}$

Statement: the group of complex automorphisms of the moduli space $M_{0,n}$ of complex $n$-marked genus 0 curves is isomorphic to $\mathfrak S_n$: one has ${\rm Aut}(M_{0,n})=\mathfrak S_n$
I believe ...

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### Behavior of the period map at the boundary of the Teichmuller space

Fix a compact oriented surface $S$ of genus $g$. Any complex structure $J$ on $S$ gives by the Hodge decomposition a linear complex structure $J'$ on $H_1(S,\mathbb{R})$. The map $J\mapsto J'$ is a ...

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### Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli

In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...

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### What is the topology of the Hitchin component?

Let $S$ be a closed orientable connected surface of genus $g\geq2$.
In Lie groups and Teichmüller space Hitchin shows that one of the component of $\chi(S)=\operatorname{Hom}^{+}(\pi_1(S),\...

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### Fenchel-Nielsen length-length coordinates on Teichmueller space?

Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...

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### What do the components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ look like?

Let $S$ be a closed orientable surface of genus at least $2$. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least $3$.
I know ...

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### Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...

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### Integration à la Mirzakhani

Let $$
\gamma = \sum_i c_i \gamma_i
$$
be a multi-curve on a hyperbolic surface $S$. For any $f: \mathbb{R}^+ \to \mathbb{R}^+$ one can define $$
f_\gamma (X) = \sum_{\alpha \in \mathrm{Mod} . \gamma} ...

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### Two definitions of Teichmüller space: relative isotopy or not?

The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism.
The definition on ...

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### Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

According to Riemann surfaces, dynamics and
geometry
by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by
$$
\|\phi\|_p = \left(\...

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### Beurling’s extremality criterion for curves: two versions

I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below,
$$
\ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|.
$$
"Extremal&...

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### Non commutative Teichmuller theory

Perhaps the first example in Teichmuller theory is the following proposition:
Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \...

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### Learning roadmap for Lorentzian geometry

I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF.
I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...

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### Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall ...

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### References on Hyperbolic Geometry and Teichmuller Theory

I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...

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### References on Riemann surfaces

I have asked the question in MSE, but did not get an answer.
I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...

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### Clarifications about a proof of (the measurable Riemann) mapping theorem in Hubbard's book on Teichmuller theory,

On page 151 of Hubbard's book, the author is proving the following theorem( Prop.4.6.2 ):
Suppose $\mu$ is a real analytic function on a domain $U$ of $\mathbb{C}$. Then every $z \in U$ has a ...

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### Relating different parametrizations of moduli space of Riemann surfaces

I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related:
On the one hand, there is a parametrization coming from hyperbolic ...

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### Thurston measure of Dehn-Thurston ball center at a multi curve

Given a surface of genus $g$ with $n$ singularities, and a decompostion of pair of pant $P=(p_1,...,p_{3g-3+n})$ one can give coordinate (called Dehn-Thurston coordinate) on the space of lamination $\...

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### Why a Teichmuller map is not a pseudo-anosov?

Let $X$ be a riemannian surface. Suppose $f:X\to X$ is a Teihmuller map with respect to a quadratic differential $q$ on $X$. This means that, if $q=dz^2$ in local coordinates in a neighborhood of ...

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### flips on labelled fatgraphs and mapping classes

A fatgraph $G$ is a graph with a cyclic ordering of the edges at each vertex. A labelled fatgraph $(G,L)$ is a fatgraph together with a labelling $L$ of each edge. A labelled fatgraph spine $(G,L,e)$ ...

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### Representation of the mapping class group in terms of flips on triangulations

$\DeclareMathOperator{\MCG}{\operatorname{MCG}}$Consider a bordered, punctured, orientable surface $S$. Associated to it there is its mapping class group $\MCG(S)$. One way to concretely think about ...

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### Harmonic maps versus Teichmuller maps between Riemann surfaces

Let $(X,\phi)$ be an element of Teichmuller space $\cal T_g$ and $q$ a (holomorphic) quadratic differential on $X$. Teichmuller geodesic flow gives a family of marked Riemann surfaces $(Y_t,\psi_t) = \...

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### When do the lengths of simple closed curves determine a hyperbolic surface?

Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...

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### "Convergence" of bordered Riemann surfaces to a congruence surface

Let $\Gamma(N)$ be the principal congruence subgroup of level $N\geq 3$, $H$ the upper half-plane and $C(N)=H/\Gamma(N)$ be the corresponding Riemann surface. In his paper " Congruence Subgroups ...

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### Well-definedness of marking a Riemann surface by diffeomorphisms in the context of Teichmüller spaces

In "An introduction to Teichmüler Theory" of Yoichi Imayoshi and Masahiko Taniguchi the Teichmüller space is defined as follows: fix a compact Riemann surface $R$ of genus $g$, a marking on ...

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### Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?

Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...

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### Weil-Petersson metric with respect to covering

Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\...

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### Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider:
The ...

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### Number of Fuchsian groups with same trace field

Let $\Gamma,\Sigma\subset \mathrm{SL}_2({\mathbb R})$ be cocompact arithmetic subgroups. They are called commensurable in the wider sense, if there exists
$g\in \mathrm{SL}_2({\mathbb R})$, such that ...

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### Holomorphic maps on moduli space and Deformation theory

Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map
$$f:\mathcal{M}\rightarrow \mathcal{F}$$
means that for each ...

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### The (measurable) Riemann mapping theorem

The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.
The measurable Riemann mapping theorem asserts the existence and ...

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### Isomorphism between two families of curves over the Teichmueller space

In his construction of the Teichmueller space of curves of genus $\geq 2$ Grothendieck states in Corollaire 2.4 that the map $$\underline{Isom}_S(X,Y) \xrightarrow{} S$$ is finite. The map represents ...