The teichmuller-theory tag has no usage guidance.

**10**

votes

**4**answers

915 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 ...

**3**

votes

**2**answers

154 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) ...

**16**

votes

**2**answers

850 views

### 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 ...

**12**

votes

**1**answer

3k 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 ...

**14**

votes

**7**answers

2k views

### 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 ...

**0**

votes

**0**answers

188 views

### Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...

**3**

votes

**1**answer

301 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 ...

**0**

votes

**0**answers

194 views

### Quick references/sources for the hyperbolic Riemann Surfaces with boundary

Hello,
Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...

**0**

votes

**1**answer

199 views

### Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$

Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...

**14**

votes

**2**answers

421 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 ...

**0**

votes

**1**answer

218 views

### Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving ...

**5**

votes

**2**answers

1k views

### 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 ...

**0**

votes

**0**answers

132 views

### Get $H^1(S,sl(2,R)_{Ad\phi}$) dimension directly from differential forms

For a genus g surface S with fundamental group $\pi$, consider Teichmuller space $Hom(\pi,SL(2,R))/SL(2,R)$, we identify tangent space at point $\phi\in Hom(\pi,SL(2,R))/SL(2,R)$ as ...

**0**

votes

**2**answers

456 views

### Unique Beltrami Differential of the form $k\frac{\bar{q}}{q}$?

I'm having a brain freeze.
Let $B$ be the space of complex valued measurable functions on the unit disk in the complex plane with essential supremum less than 1. Then, the universal Teichmuller space ...

**2**

votes

**2**answers

504 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 ...

**4**

votes

**3**answers

442 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 ...

**4**

votes

**1**answer

345 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 ...

**2**

votes

**1**answer

526 views

### 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 ...

**2**

votes

**1**answer

987 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) ] ...

**3**

votes

**0**answers

381 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 ...

**5**

votes

**2**answers

301 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 ...

**2**

votes

**0**answers

184 views

### 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 ...

**3**

votes

**1**answer

495 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 ...

**1**

vote

**0**answers

228 views

### A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...

**2**

votes

**0**answers

149 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) = ...

**5**

votes

**1**answer

166 views

### 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 ...

**1**

vote

**1**answer

372 views

### 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 $d_X,d_Y$ ...

**4**

votes

**1**answer

608 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 ...

**1**

vote

**3**answers

469 views

### 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 ...

**2**

votes

**1**answer

172 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 ...

**4**

votes

**1**answer

375 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 ...

**3**

votes

**1**answer

2k 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 ...

**2**

votes

**0**answers

353 views

### Calculation of dimension of holomorphic quadratic differentials as in Gardiners book

In Frederick Gardiner's book Teichmuller Theory and Quadratic Differentials, P.27-28, Chapter 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n $ ( by using Riemann-Roch theorem ). Now for open annulus ...

**3**

votes

**1**answer

704 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 ...

**0**

votes

**2**answers

700 views

### A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

Hello,
I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...

**1**

vote

**1**answer

463 views

### Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello ,
I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I ...

**1**

vote

**1**answer

671 views

### 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 ...

**2**

votes

**1**answer

185 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 ...

**1**

vote

**1**answer

311 views

### 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 ...