The teichmuller-theory tag has no wiki summary.

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### Explicit homeomorphism between Thurston's compactification of Teichmuller space and the closed disc

Thurston's celebrated compactification of Teichmuller space was first described in his famous Bulletin paper. Teichmuller space is notorously homeomorphic to an open disc of some dimension (this can ...

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### Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?

Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...

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### What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:
No-wandering-domain Theorem (Sullivan)
Theorems of Thurston ...

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**4**answers

882 views

### Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference
(but he remembers seeing them in the literature):
Let $f$ be a holomorphic function mapping the ...

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**1**answer

2k views

### What is “Teichmüller Theory” and its history ?

What is "Teichmüller Theory" ? What part has been worked out / forseen by O. Teichmüller himself and what is further development ? Is there some current work which might be considered as ...

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187 views

### Comparing different layered structures for fibered 3-manifolds: example request.

Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...

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**1**answer

260 views

### For which surfaces is Penner's conjecture known to be true?

Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...

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### Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$.
The ...

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### What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with ...

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273 views

### Optimal pants decompositions of a hyperbolic surface

Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$.
Here is a candidate ...

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### Identification of conformal classes of pos def quadratic forms on R^2 with unit ball

One of the lemmas at the foundation of Teichmuller theory is as follows. Let $Q(x,y)$ be a positive definite quadratic form. Then there exists unique $\lambda \in \mathbb{R}$ and $\mu \in ...

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554 views

### A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U ...

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**1**answer

337 views

### Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we ...

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**1**answer

183 views

### Link for “A spine for Teichmüller space”, preprint by Thurston

Can someone please give any link or mention any source where I can find the following preprint.
W.Thurston, A spine for Teichmüller space, preprint, three pages, 1986.

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**1**answer

1k views

### Why use Teichmuller representatives?

In p-adic mathematics, what is the advantage of using Teichmuller representatives over using just the numbers 0,1,2,...,p-1 ?
In either case, the norm is the same.
In either case, all the points are ...

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402 views

### Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ...

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327 views

### Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps

If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get ...

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52 views

### Distorsion of subgroups of the mapping class group

Let $S_{g,b}$ be an oriented surface with $b$ boundary components and $S_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S_{g,b})$ and $\mathrm{Mod}(S_g^b)$ their (orientation ...

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109 views

### (Un)distorted subgroups in the mapping class group: reference required.

Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results:
the cyclic subgroup generated by a pseudo-Anosov element ...

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116 views

### Euclidean surfaces with conical singularities and cusped hyperbolic surfaces

Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$.
Construction 1: it is well-known that the conformal class ...

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108 views

### continuity of length function $l: T(X) \times MF \to \mathbb R$

Let $T(X)$ be the Teichmuller space of a closed Riemann surface $X$ of genus $g \geq 2,$ and $MF$ the space of equivalence classes of measured foliations. Then we have a length function
$l: T(X) ...

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272 views

### Strata of quadratic differentials from rational billiards

Given a quadratic differential $q$ on a surface of genus $g$, we say that $q\in \mathcal Q(k_1,\ldots,k_n)$ if $q$ has $n$ distinct zeroes of order $k_1,\ldots,k_n$ respectively. The set $\mathcal ...

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618 views

### Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...

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### $\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...

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367 views

### Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:
Let us consider the orientation-preserving homeomorphic solutions $f: D ...

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**1**answer

692 views

### How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...

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### Hyperbolic structures on once punctured tori

I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...

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### Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site.
Let $S$ be some orientable surface obtained by removing ...

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284 views

### teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...

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### iwaniec's conjecture

Does anyone know whether there is any geometric applications of the iwaniec's conjecture on $ l^p $ bound of beurling alfhors transform( or the complex hilbert transform). One application could have ...

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### The version of Montel's theorem used in the proof of Jenkins-Strebel differential

Hello,
I am afraid that my main question might be a bit too elementary, but still I ask :
In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...

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177 views

### Defining the natural co-ordinate for a holomorphic quadratic differential near a zero of odd order

I was reading a book [ Teichmuller Theory and quadratic differential and Farb-Margalits' A Primer on MCG ] where they define the natural co-ordinate of holomorphic quadratic differential on a compact ...

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### Coordinates for Teichmuller space for compact conformal surfaces

Fenchel-Nielsen coordinates give a coordinatization of Teichmuller space for compact conformal surfaces admitting a pants decomposition. But not all compact conformal surfaces (possibly with boundary, ...

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### A continuous version of Teichmuller uniqueness

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and ...

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161 views

### Coefficients of lacunary series on quasiconformally transformed unit disk

Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if ...

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### Explicit local expression for Bers embedding in genus 2

Let $\mathcal T_{g,n}$ be the Teichmüller space of genus g compact Riemann surfaces with $n$ marked points. According to Riemann, this is a complex manifold of complex dimension 3g-3+n.
Bers ...

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### The space of conformal structures on the Euclidean plane

Since $\mathbb R^2$ is a non-hyperbolic surface, it is ignored in most accounts of Teichmuller theory. I look for papers where the space of conformal structures on $\mathbb R^2$, and the associated ...

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146 views

### Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...

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474 views

### Does normalized Ricci flow on surfaces yield a bundle?

As is well known,
the normalized Ricci flow is defined for all $t>0$ on compact surfaces,
and every metric on a compact surfaces converges to a metric constant curvature if $X \neq S^2$ (at least I ...

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### How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets

$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...

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### About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps:
If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$:
...

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### Quasi-isometric embeddings of the mapping class group into the Teichmuller space

Does there exist a quasi-isometric embedding
$$MCG(S) \to (\mathrm{Teich}(S), d)$$
for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?

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### Is there an Error on pg. 17 of Tromba's “Teichmuller Theory in Riemannian Geometry”?

I'm pretty sure that this is a minor error, but I could use some help here. So the book I'm referring to in the title is this book (MR1164870).
On pg. 16-17, he is proving that the space of almost ...

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### Does the Teichmüller space of the pair of pants admit a continuous global section?

Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth ...

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### Ratner theorem and dense geodesic planes in hyperbolic manifolds

Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...

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### Centralizer of a pseudo-Anosov element

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?

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### Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states :
Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...

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### Conjugate Groups of (quasi) Fuchsian Groups

I apologize in advance if this question is so trivial or too low level.
Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...

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### Length of the transversal for surfaces with cusps

In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a ...

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211 views

### References for Teichmüller space of pointed elliptic curves

Where can I find an elementary introduction (construction, description, main properties) to the Teichmüller space ${\cal T}_{1,n}$ of elliptic curves with $n$-marked points?
Same question for the ...