Skip to main content

Questions tagged [teichmuller-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
9 votes
0 answers
100 views

Symplectic form on the space of geodesic currents on a surface?

There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...
Ian Agol's user avatar
  • 67.4k
0 votes
1 answer
136 views

Teichmüller theory for open surfaces?

I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces? My motivation basically is that I would like to find out more about the "...
M.G.'s user avatar
  • 6,730
4 votes
1 answer
239 views

Reference for Teichmuller spaces of punctured surfaces

What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$? I am looking for a reference where there is the correct statement and or proof of say the Bers embedding,...
Chitrabhanu's user avatar
3 votes
1 answer
124 views

The boundary regularity of a Teichmüller domain

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
2 votes
1 answer
91 views

Parameterizing Teichmüller spaces of punctured surfaces

Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
Yousuf Soliman's user avatar
2 votes
1 answer
172 views

Example of pseudo $3$-manifold without any shape structure

I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me: ...
Shana's user avatar
  • 237
7 votes
1 answer
465 views

Goldman symplectic form vs Weil–Petersson symplectic form

I'm confused about the exact multiplicative factor that relates Goldman symplectic form on the $\operatorname{SL}(2,\mathbb R)$-character variety and the Weil–Petersson symplectic form on Teichmüller ...
AMath91's user avatar
  • 38
0 votes
1 answer
195 views

Fixed points free automorphisms of Teichmüller spaces

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
Mahdi Teymuri Garakani's user avatar
2 votes
1 answer
226 views

An interior cone condition for Teichmuller spaces

Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
Mahdi Teymuri Garakani's user avatar
7 votes
1 answer
839 views

Why is the length spectrum called a spectrum?

Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$. Question: is $\mathcal{L}(X)$ a ...
Andrey Ryabichev's user avatar
1 vote
0 answers
54 views

Comparison between two volume forms on genus zero Teichmüller space

Consider a sphere with $n$ punctures. If you pick a holomorphic cotangent vector at each puncture, you can canonically construct a holomorphic top form in the corresponding moduli space. (The specific ...
Charles Wang's user avatar
1 vote
1 answer
48 views

Bound on the sum of intersection number of any projectivized measured foliation with two transverse measured foliations

Let $R$ be a finite Riemann surface (having negative Euler Characteristic) without boundary (may have punctures) and $q$ be a unit area quadratic differential on $R$. We define $\mathcal{MF}_{1}=\{F \...
P.S's user avatar
  • 221
2 votes
1 answer
210 views

Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism

This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand: $$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
Kenny S's user avatar
  • 77
3 votes
1 answer
296 views

Representation theory and topology of Teichmüller space

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\char{char}$I am reading a note on Teichmüller space, and I come across a somewhat algebraic problem in the picture below,...
Kenny S's user avatar
  • 77
5 votes
1 answer
363 views

Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title: The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
Chaitanya Tappu's user avatar
0 votes
1 answer
115 views

Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds

In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as $$g = \frac{...
JMK's user avatar
  • 299
1 vote
1 answer
90 views

Common holomorphic forms for two distinct complex structures

Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
Dorian's user avatar
  • 331
2 votes
0 answers
81 views

Two different Bers embeddings

In An Introduction to Teichmüller spaces by Imayoshi and Taniguchi, they present in section 6.1.3 the Bers embedding as a map from Teichmüller space of a Riemann surface $X$ to the space of quadratic ...
Jacques's user avatar
  • 563
1 vote
0 answers
68 views

Local Chart for Teichmuller Space as A Manifold

Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem,...
CharlieHo's user avatar
1 vote
0 answers
80 views

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 ...
A B's user avatar
  • 99
2 votes
1 answer
162 views

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 ...
Leo Moos's user avatar
  • 4,968
4 votes
0 answers
154 views

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 $\mathcal{J}(\Sigma)$ ...
Joaquin Lema's user avatar
2 votes
1 answer
193 views

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}\...
Yuxiao Xie's user avatar
2 votes
0 answers
175 views

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 ...
RegularGraph's user avatar
1 vote
0 answers
64 views

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; ...
TSBH's user avatar
  • 121
5 votes
0 answers
161 views

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(\...
Atlas Tasilli's user avatar
2 votes
0 answers
105 views

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 ...
user avatar
2 votes
0 answers
282 views

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 ...
Josh Lam's user avatar
  • 222
62 votes
6 answers
48k views

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 ...
Madeleine Birchfield's user avatar
2 votes
0 answers
121 views

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 ...
Sidharth Ghoshal's user avatar
2 votes
1 answer
144 views

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} \...
P.S's user avatar
  • 221
1 vote
0 answers
243 views

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 ...
Sidharth Ghoshal's user avatar
5 votes
1 answer
254 views

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\...
David.D's user avatar
  • 423
3 votes
1 answer
251 views

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 ...
giulio bullsaver's user avatar
4 votes
2 answers
219 views

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 ...
Harry Reed's user avatar
7 votes
0 answers
158 views

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 ...
 V. Rogov's user avatar
  • 1,160
2 votes
0 answers
83 views

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 ...
Lucien's user avatar
  • 828
3 votes
0 answers
100 views

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 ...
Julien Marché's user avatar
2 votes
0 answers
71 views

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 ...
Thomas Kojar's user avatar
  • 4,459
1 vote
0 answers
317 views

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),\...
Jacques's user avatar
  • 563
5 votes
2 answers
431 views

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 ...
JHM's user avatar
  • 2,274
15 votes
2 answers
842 views

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 ...
Jacques's user avatar
  • 563
7 votes
3 answers
331 views

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 ...
Lisa's user avatar
  • 71
7 votes
0 answers
328 views

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} ...
EtienneBfx's user avatar
2 votes
1 answer
221 views

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 ...
Ma Joad's user avatar
  • 1,673
1 vote
1 answer
304 views

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(\...
Ma Joad's user avatar
  • 1,673
2 votes
0 answers
60 views

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&...
Ma Joad's user avatar
  • 1,673
3 votes
0 answers
185 views

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 \...
Ali Taghavi's user avatar
3 votes
2 answers
363 views

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 (...
user2022's user avatar
7 votes
1 answer
382 views

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 ...
aglearner's user avatar
  • 14k

1
2 3 4 5 6