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### 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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### 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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275 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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168 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 ...

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### 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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### 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 ...

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### 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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### 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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### 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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### 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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441 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 ...

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163 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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### 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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### 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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288 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 ...

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### 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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656 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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### 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 ...

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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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### 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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278 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 ...