The teichmuller-theory tag has no usage guidance.

**16**

votes

**2**answers

945 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 ...

**15**

votes

**7**answers

3k 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 (...

**14**

votes

**2**answers

440 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 ...

**14**

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 continuation/...

**14**

votes

**2**answers

2k views

### 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 ...

**14**

votes

**1**answer

331 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

I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...

**10**

votes

**4**answers

934 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 ...

**9**

votes

**1**answer

527 views

### A question about Mirzakhani et. al.'s algebraicity theorem

While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...

**8**

votes

**2**answers

159 views

### Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.
Suppose that we have closed surface of genus $g\geq 2$, and ...

**7**

votes

**2**answers

237 views

### Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties:
For every two points in the plane there exists a unique geodesic joining them.
Every geodesic determines exactly two points on the ...

**6**

votes

**2**answers

234 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 ...

**6**

votes

**1**answer

160 views

### Geometric quantization of Teichmuller space

The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric ...

**5**

votes

**3**answers

832 views

### 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 ...

**5**

votes

**1**answer

278 views

### Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture:
Every object naturally associated to a surface S and having
a sufficiently rich structure has $Mod(S)$ as its groups of ...

**5**

votes

**2**answers

2k 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 ...

**5**

votes

**2**answers

306 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 ...

**5**

votes

**1**answer

118 views

### Metrics on Teichmüller spaces

I know that Teichmüller $\mathcal{T}_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other ...

**5**

votes

**2**answers

111 views

### Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface.
Definition: A Quasiconformal surface $S$ is a ...

**5**

votes

**1**answer

261 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.

**5**

votes

**1**answer

167 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 $\mu \in \mathbb{C}$...

**5**

votes

**1**answer

188 views

### Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...

**4**

votes

**3**answers

464 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

625 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 \...

**4**

votes

**1**answer

234 views

### Angle between geodesics in hyperbolic surface

Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...

**4**

votes

**1**answer

156 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 ...

**4**

votes

**1**answer

393 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 ...

**4**

votes

**1**answer

194 views

### Canonical models of Teichmüller curves

It is well-known that Shimura varieties can be defined over number fields and that moreover they possess canonical model over number fields. On the other hand, Teichmüller curves can also be defined ...

**4**

votes

**1**answer

122 views

### $L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} &...

**4**

votes

**1**answer

159 views

### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...

**4**

votes

**1**answer

352 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 ...

**4**

votes

**1**answer

127 views

### Definition of the Teichmuller space via topological marking of the $\pi_1$

Let $S_0$ be a compact orientable and oriented surface, of genus $g$, fixed once for all. For a fixed point $s_0\in S_0$, one considers a symplectic basis $a_1,\ldots,a_g,b_1,\ldots,b_g$ of $\pi_1(...

**4**

votes

**1**answer

233 views

### The hyperbolic metric on a flat surface

Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$)...

**4**

votes

**0**answers

146 views

### Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...

**4**

votes

**0**answers

86 views

### Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...

**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 ...

**3**

votes

**1**answer

194 views

### A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...

**3**

votes

**1**answer

533 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 :H^...

**3**

votes

**1**answer

89 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 ...

**3**

votes

**2**answers

451 views

### 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 ...

**3**

votes

**1**answer

153 views

### Decomposition of hyperbolic surfaces near cusps into annuli

Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near ...

**3**

votes

**1**answer

188 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 $[g]$...

**3**

votes

**2**answers

166 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) \...

**3**

votes

**1**answer

313 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 Q(...

**3**

votes

**1**answer

732 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 ...

**3**

votes

**1**answer

366 views

### Confusion about a result on Shimura and Teichmüller curves

It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by $y^4=x(x-1)(...

**3**

votes

**0**answers

123 views

### Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity.
Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...

**3**

votes

**0**answers

143 views

### $\mathbb{CP}^1$-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 ...

**3**

votes

**0**answers

389 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 \...

**2**

votes

**2**answers

512 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 ...

**2**

votes

**1**answer

1k 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) ] ;...