Questions tagged [teichmueller-theory]

Questions related to the work and continuation of Oswald Teichmüller on Teichmüller theory, especially Teichmüller spaces.

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17 views

Induced map from the universal cover of the base space to the Teichmuller space of the fiber is holomorphic

Let $\phi:X\to Y$ be a submersion and holomorphic map with not every two fibers are biholomorphic, where $X$ a compact complex surface, $Y$ is a Riemann surface. Let $\pi:U\to Y$ be the universal ...
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31 views

how do the Lp spaces sit inside Teichmüller space?

I've heard it said that Teichmüller space gives a metric to the space of all metric spaces. If this is so, where do the Lp spaces sit in the space of all metric spaces?
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30 views

Homeomorphism Between Closed Riemann Surfaces Homotopic to Quasiconformal Mapping

I'm re-reading a paper of Bers and for the second time, and I am yet again confused about the claim in the title, which Bers declares to be easy to prove. For context, I'll lay out some terminology....
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30 views

Atlas for hyperbolic pair of pants

I was reading about Teichmüller spaces and how you give a hyperbolic structure to a pair of pants via glueing two right angled hexagons, but I wanted to know if there was an explicit way to describe ...
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114 views

A Third Derivative in Ahlfors' `Some Remarks on Teichmuller Space'

I'm slugging my way through Ahlfors' "Some Remarks on Teichmuller Space" and am stuck on the calculation of equation (1.18). The problem here is to compute the third derivative, on the unit disk $\...
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1answer
18 views

Surjectivity of parameters for torus double cover of sphere

Let $T=\mathbb{C}/\Lambda$ be a torus, viewed as a Riemann surface. Quotienting out by the relation $z\sim -z$ gives a double cover $\pi : T \to \hat{\mathbb{C}}$ of the Riemann sphere, ramified over ...
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1answer
165 views

Understanding Teichmuller equivalence of marked compact Riemann surfaces

Let me give some definitions to begin with. Let $\Lambda$ be a compact orientable two-dimensional (over $\mathbb R$) manifold of genus $p>2$. Let $g$ be a conformal structure on $\Lambda$, and $...
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1answer
101 views

Could someone explain to me what a Teichmuller Space is?

In the simplest terms possible, for someone who understands the basics of Manifolds, Topology, but barely any of the more complicated topics. I've been using the following: http://homeowmorphism.com/...
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70 views

Moduli Space of Tori identified with $\mathbb{C}$

I am currently reading through the book An Introduction to Teichmüller Spaces by Imayoshi and Taniguchi. In Section 1.2, we see that $M_1$, the moduli space of tori, can be identified with $\mathbb{H}/...
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1answer
106 views

Why is the matrix representation of an almost complex structure like this?

Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
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57 views

Method of characteristics for the Beltrami equation when $\mu$ is real analytic

I am reading a proof of the measurable mapping theorem which shows the existence of a solution to the Beltrami equation in the simple case when $\mu \in L^{\infty}(\mathbb{C})$ is real analytic (due ...
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1answer
65 views

Any multiplicative character $χ: \mathbb{Z}/p\mathbb{Z} \to Z_p$ is a power of the Teichmuller character

This is a follow-up question to Existence/uniqueness of the Teichmuller map for p-adic integers Let $Z_p$ denote the p-adic integers. Let $π:Z_p \to \mathbb{Z}/p\mathbb{Z}$ by $π(a_0+a_1p+a_2p^2+...)=...
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1answer
45 views

Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...
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74 views

About a proof on Hubbard's Teichmüller Theory

I'm reading the book "Teichmüller Theory and Applications to Geometry, Topology and Dynamics" by John Hubbard and I have the following question. The proposition 7.4.4 says: Let $\varphi,\psi$ be ...
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3k views

Why does Mochizuki insist on “forgetting the previous history of an object”?

What is the simplest (at the lowest level feasible) explanation of the approach of “forgetting the history” of a mathematical object, as used in Inter-universal Teichmüller Theory (IUTT)? Please ...
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1answer
85 views

Bubbling off in Deligne-Mumford compactification

I'm trying to understand an argument in Casim Abbas' 'An Introduction to Compactness Results in Symplectic Field Theory'. Here's where I stuck: (It's on chapter 3.3.2, Adding additional marked points ...
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1answer
114 views

Mapping Class Group acts properly discontinuous; Alexander method

Let $S$ be a closed surface of genus $g$. The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" (p.59) (roughly) states that if $c_1,c_2$ are two filling curves in minimal ...
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341 views

A good reference to learn Teichmuller Theory? [closed]

I'm looking for a reference to start learning Teichmuller Theory.
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186 views

Teichmuller disk and $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...
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1answer
67 views

Fixed points of finite order mapping classes

I am currently working myself through the proof that every finite order element of the mapping class group $\mathrm{MCG}(S_g)=\mathrm{Hom}^+(S_g) / \mathrm{Hom}_0(S_g)$ of a closed, hyperbolic, genus $...
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1answer
65 views

About partial derivatives

I'm reading the book "Teichmüller Theory and Applications to Geometry, Topology and Dynamics" by Hubbard and I have the following problem. I need to compute $$\frac{d}{ds}\frac{\overline{\partial}z+\...
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1answer
482 views

How to interpret the space of quadratic differentials as a cotangent space?

The Wikipedia article "Quadratic Differential" opens with the following text: In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic ...
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48 views

Reference for convex norm ball in space of quadratic differential

I have heard in a few talks that the unit norm ball on the cotangent space of Teichmüller space is convex. I am looking for a reference where that statement would be proved. I am also looking for a ...
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Parametrization of Teichmüller space

I'm trying to learn Teichmüller theory, but appear to get stuck early on. Let $\Sigma$ be a smooth closed oriented surface of genus $g\geqslant 2$ and let $\mathrm{Conf}(\Sigma)$ denote the set of ...
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1answer
50 views

Equivalent markings on Riemann surfaces

This is the definition in Introduction to Teichmuller Spaces by Imayoshi. Let $(R,\Sigma_p)$ be a pair where $R$ is the riemann surface and $\Sigma_p$ is called marking where $\Sigma_p$ is a set of ...
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1answer
51 views

Checking that a function is in $L^p(\mathbb{C})$

I've been reading Imayoshi & Taniguchi's text "An Introduction to Teichmuller Spaces" and while going through the proof of Lemma 4.20 realized that I'm not comfortable integrating functions over ...
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1answer
131 views

Uniformly convergent sequence of quasiconformal mappings

Let $G \subseteq \mathbb{C}$ be a domain (one may assume that $G$ is bounded and simply connected, if needed). Suppose $(f_n)_{n \in \mathbb{N}}$ is a sequence of bounded (i.e. $\sup_{z \in G} |f_n(z)|...
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98 views

Formula for the Length Function on Teichmuller Space

Let $S_g$ denote the closed orientable surface of genus $g\geq 2$. Then there is a natural bijection between the Teichmuller space $\text{Teich}(S_g)$ of $S_g$ and the set of all the discrete-faithful ...
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75 views

Teichmuller space as Discrete Faithful Representations up to Conjugation

Let $S_g$ denote the closed orientable surface with genus $g\geq 2$. A marking of $S_g$ is a diffeomorphism $\phi:S_g\to X$, where $X$ carries a hyperbolic metric on it. Two markings $\phi:S_g\to X$ ...
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1answer
120 views

About a definition of marking equivalence

I'm reading chapter 6 about Teichmüller space in the book: Geometry and Spectra of Compact Riemann Surfaces by Peter Buser. In the following definition: Definition 6.1.2: Two marked Riemann ...
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1answer
173 views

Different Definitions of Teichmuller Space

Let $S$ be a compact smooth surface. On pg 276 of Farb and Margalit's A Primer on Mapping Class Groups, the following definition of the Teichmuller space of $S$ is given. A hyperbolic sturcture on $...
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1answer
71 views

Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

Assume that $\mathcal{R}$ is a hyperbolic surface with $m$ geodesic boundary components and $n$ punctures. If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is ...
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439 views

Hyperbolic Metric on a Riemann Surface

From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the ...
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64 views

A model for conformal structures on polygon

By a conformal structure on a polygon, I mean all the vertices of a polygon are missing (or, equivalently, it has ideal vertices), so that it can be considered as a conformal structure on the unit ...
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1answer
847 views

Is there any point in reading about Inter-universal Teichmüller theory (unless one works on it)?

Is there any point in reading about Inter-universal Teichmüller theory (unless one works on it)? Does IUT have any applications known yet? I'm mainly interested in it for intellectual curiosity ...
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44 views

How to show this mapping is quasiconformal. And the integrability of the gradient.

A complex map $f$ on the unit disk defined as $f(re^{i\theta})=r^ke^{i\theta}$ where $k>1$. I hope to know how to show this map is quasiconformal and what is the largest $p$ such that the gradient ...
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1answer
142 views

Question on Teichmuller metric as infimum of the dilatations

I'm learning Teichmuller theory from the book "Primer on Mapping class group". There is something I can not understand, I hope someone could clarify it. Let $\mathcal{T}_g$ be the Teichmuller space ...
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204 views

orientation$−$ preserving or orientation$-$reversing in the Definition of Teichmuller space

There are several questions about the definition of Teichmuller space. Definition(see A Primer on Mapping Class Groups) Let $S$ be a genus g closed surface with $g \ge 2$.The Teichmuller space of $S$...
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1answer
547 views

How to understand an triangulation of torus?

In the paper, it is said that Figure 1.1 in Example 1.2.3 is a triangulation of a torus. How to see that every face in this triangulation is an triangle? How many triangles are there in this ...
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95 views

Does the “send-basis-to-basis isomorphisms creation process” work for fundamental groups (at least for closed orientable surfaces of genus $g$)?

Definition: A marking on a sphere $S$ with $g$ handles ($g<\infty$) is an ordered collection $\Sigma_p=\{[\alpha_j],[\beta_j]\}_{j=1}^g$ of elements $[\alpha_j],[\beta_j]$ all of them in a same ...
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1answer
68 views

For all orientation-preserving homeo is there a homotopic orientation-preserving diffeo?

Basically, I want to know if the following is true: Given an orientantion-preserving homeomorphism $f:R\to S$ between Riemann surfaces $R$ and $S$, does exist an orientation-preserving ...
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1answer
399 views

How to interpret analogy of Gaussian integral of Alien Copies in IUT theory by Mochizuki?

In Mochizuki's paper, he refers to copy of Integral as 'Alien Copy', which is strange since the double integral is an extension of the surface of uni-dimensionnal integral due to Summation properties (...
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1answer
249 views

Relation between moduli of curves and Teichmüller theory

Both the theory of moduli of curves and Teichmüller theory seem to be concerned with the moduli of Riemann surfaces. However, they appear to belong to different fields within mathematics. Could ...
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1answer
89 views

How is the trace field of a hyperbolic surface defined?

I am familiar with the construction and general properties of trace fields for hyperbolic $3$-manifolds. But in that setting we use Moscow-Prasad rigidity to define these fields as manifold invariants....
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290 views

Why are there no holomorphic quadratic differentials on a complex torus?

It is well-known that the space or strata $\mathcal{Q}(\emptyset)$ of quadratic differentials with no poles or zeros on a genus 1 surface is empty (this is one of the four exceptions to a theorem of ...
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1answer
175 views

Homotopy in Teichmüller space definition: to be or not to be? That is the question

In the book Introduction to Teichmüller Spaces, by Imayoshi & Taniguchi, we finde the following definition of the Teichmüller space of a Riemann surface $R$, denoted $T(R)$: I want to draw ...
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2answers
450 views

Every Riemann Surface has a countable basis for its topology

In the book Introduction to Teichmüller Spaces, by Taniguchi and Imayoshi, we have the following definition for a Riemann Surface: At the following pages, the authors make a remark recalling some of ...
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1answer
111 views

Dehn twist and length of a curve.

Let $S$ be a closed hyperbolic surface and $\alpha$ be a simple closed curve. Let $T_{\alpha}$ be the left Dehn twist along $\alpha$. Let $l_x$ denote the length of the geodesic representative in the ...
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1answer
103 views

$f$ conformal $\implies$ $f$ ACL?

In An Introduction to Teichmüller Spaces, by Imayoshi and Taniguchi, we have the following definition for an absolutely continuous function: where, I suppose, a function $g:I\to \mathbb{C}$, $I \...
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262 views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be ...