Questions tagged [teichmueller-theory]

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

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Different definitions for the Teichmüller space of puctured spheres

My question is about understanding of equivalence of two different definition for the Teichmüller space of hyperbolic surfaces $\mathbb{S}^2 \setminus P$, where $|P| \geq 3$. The first definition for ...
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25 views

Question about Hubbard's analytic definition of quasiconformality. Aren't weak derivatives only defined up to a set of measure zero?

I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/...
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1answer
42 views

Teichmuller space of the 4-punctured sphere

I'm a bit confused working with Teichmuller space at the moment. Let's think of Teichmuller space as the space of holomorphic/conformal structures on a surface, up to diffeomorphisms isotopic to the ...
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55 views

When is a g-dimensional subspace of $H^1(S; \mathbb{C})$ the $H^{1,0}$ of a complex structure on $S$?

Given a closed surface $S$ of genus $g \geq 1$ there are lots of choices of complex structure, and each one singles out a subspace of the surface's first cohomology group with complex coefficients ...
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1answer
47 views

Automorphism classes of branched covers of disks over disks

I was reading Hubbard's book (Teichmuller Theory Vol 2) and in a proof (9.3.2), he mentions that there is one branched cover of the disk over a disk with 1 ramification point (degree $k$) up to ...
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65 views

Learning Roadmap for 3-dimensional Hyperbolic Geometry

I am interested in learning Teichmuller theory. Previously, I learnt 2 dimensional hyperbolic geometry from S. Katok's book. Now, I am interested in learning about significance of hyperbolic geometry ...
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48 views

Quotient of hyperbolic disc model

I am trying to learn decorated Teichmüller theory and reading Penner's "Decorated Teichmüller Theory" book. In Chapter 2 in "Punctured Surfaces" section I got confused. Let $\...
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1answer
189 views

What is Representation of Surface Groups?

I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
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1answer
120 views

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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68 views

Lemma 3.6 in Kerckhoff's "The Nielsen Realization Problem"

I am confused by the proof of Lemma 3.6 in Kerckhoff's "The Nielsen Realization Problem." Let $\gamma \in S$ ($S$ is the set of isotopy classes of simple closed curves) and $\bar{\gamma}(t)$ ...
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1answer
66 views

Thurston Compactification

Sorry in advance for my English. I'm studying the Thurston compactification from the Jean-Pierre Otal's book "The Hyperbolization Theorem for Fibered 3-Manifolds". I have a question, what $\mathbb{...
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1answer
87 views

When might Fenchel-Nielsen twist coordinates exceed 1/4?

When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length ...
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68 views

Singularities of moduli spaces $M_g$

This survey paper from Lizhen Ji says: Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he ...
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49 views

Fourier coefficients of a function

Prove that for $w(\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{z}-1)(\...
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34 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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66 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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87 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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1answer
76 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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2answers
120 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
134 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
202 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
228 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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234 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
270 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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73 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
220 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
70 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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106 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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1answer
4k 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
129 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
253 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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609 views

A good reference to learn Teichmuller Theory? [closed]

I'm looking for a reference to start learning Teichmuller Theory.
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1answer
297 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
133 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
67 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
1k 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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1answer
53 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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76 views

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
76 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
60 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
215 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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1answer
139 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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129 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
214 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
268 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
93 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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1answer
931 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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71 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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996 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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1answer
60 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 ...