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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 am new to …

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 …

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 $ …

In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic surface …

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, with …

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 : …

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 hype …

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 appre …

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 denot …

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 o …

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ …

Hi, my question is : Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. …

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 $d_X,d_Y$ r …

I know the definition of k -quasi-symmetric maps $f$ on $R$,it is there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$ So I just want to double c …

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 …