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175 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
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1answer
298 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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0answers
186 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 ...
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1answer
194 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 ...
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2answers
409 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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1answer
213 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
2answers
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 ...
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0answers
131 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
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2answers
407 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
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2answers
495 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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3answers
421 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
1answer
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
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1answer
501 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
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1answer
906 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
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0answers
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
2answers
294 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
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0answers
177 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
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1answer
471 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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0answers
222 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
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0answers
148 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
1answer
164 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 ...
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1answer
365 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 $d_X,d_Y$ ...
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1answer
602 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
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3answers
463 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
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1answer
169 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
1answer
364 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
1answer
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 ...
2
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0answers
338 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
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1answer
688 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
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2answers
690 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, ...
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1answer
460 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
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1answer
664 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
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1answer
183 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
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1answer
305 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 ...