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$\newcommand{\Reals}{\mathbf{R}}$There is no minimal surface of constant negative Gaussian curvature in $\Reals^{3}$, even locally. Up to scaling, the principal curvatures would satisfy $$\kappa_{2} = -\kappa_{1},\qquad -1 = \kappa_{1} \kappa_{2} = -\kappa_{1}^{2},$$ so the principal curvatures would be constant: $\kappa_{1} = 1 = -\kappa_{2}$ without ...

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Question 1: We know that from the uniformization theorem, the Riemann surface is covered by either one of these: $$\tilde M = \mathbb S^2 , \mathbb C, \ \text{or }\mathbb H.$$ Then for any Riemann surface $M$, $M$ is biholomorphic to $\tilde M/\Gamma$, where $\Gamma$ is a subgroup of bilomorphism on $\tilde M$ and each $F\in \Gamma$ has no fixed points. ...

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Here are a few references to get you started: Thurston's notes and the book form, Geometry and Topology of 3-Manifolds Introduction to Teichmuller Theory, by Hubbard Lectures on Hyperbolic Geometry, by Benedetti and Petronio

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As long as you are careful about the identification $f : PSL(2,\mathbb{R}) \to T^1(\mathbb{H})$, the answer is that the two measures are the same. Let me avoid actual calculations and list the properties that you need $f$ to verify: $f$ is smooth. $f$ is an equivariant map with respect to the two natural actions of $PSL(2,\mathbb{R})$: the natural (left) ...

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Yes! Denote by $\pi$ the covering map $\pi: \mathbb{H} \to S$. There are many lifts of $\gamma$ corresponding to the many choices of the starting point $\tilde{\gamma}(0) \in \pi^{-1}(\gamma(0))$. But after one chooses a starting point, there is a unique lift of $\gamma$ to a curve $\tilde{\gamma}$ since $\pi$ is a covering map (this follows from a general ...

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The semicircular geodesic in this model is the analog of an infinitely extended straight line in Euclidean space. Attempting to parametrize it starting at an endpoint is the same as trying to parametrize a Euclidean line starting at infinity - it doesn't really make sense. What you need to do is choose your parameter to be zero somewhere in between ($(x,y) = ... 2 Suppose that$z \in O$. We need to find$\epsilon > 0$such that$w \in O$for all$w$with$\left|w-z\right| < \epsilon$Let$z = -a + bi$, where$a,b$are positive reals. Then just take$\epsilon = \text{min}(a,b)$. For then, if$\left|w-z\right| < \epsilon$, where$w = -c + di,we have $$\left|a-c\right| + \left|b-d\right| \leq ... 2 Let z_{0}=x_{0}+iy_{0} (x_{0} and y_{0} real numbers) be in this quadrant. Consider the ball D(z_{0},r_{0}) where r_{0}=\min(|y_{0}|,|x_{0}|). Assume that z=x+iy (x,y \in \mathbb{R}) \in D(z_{0},r_{0}). Then |z-z_{0}|=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}<r_{0}. Hence |x-x_{0}|<|x_{0}| hence x_{0}<x-x_{0}<-x_{0}, [x_{0}<0 so ... 1 Let's consider an arbitrary hyperbolic triangle with angles \alpha, \beta, \gamma opposite respective sides a, b, c. The area (T) is given by$$T = \pi - \alpha - \beta - \gamma \tag{0}$$so that, writing X_2 for X/2,$$\begin{align} \cos T_2 &= \cos\left(\frac{\pi}{2}-\alpha_2-\beta_2-\gamma_2\right) = ... 1 I read one book long ago. I am not an expert in this area, but I also had came up at similar questions, and found following book interesting. In the book Complex Functions: Algebraic and Geometric Veiw-point, the last one or two chapters discuss in detail (as per my understanding) the topics you mentioned. I hope that you will find it suitable and easier to ... 1 What is the area? Well, at least we want it to be (1) a non-negative function of a polygon that is (2) additive:S(A\cup B)=S(A)+S(B)$if$A\cap B$has no interior points. It turns out, these two simple properties define$S$almost uniquely — it's unique up to multiplication by a constant. Now in Euclidean geometry one can prove that$S(\Delta)=ah_a$by ... 1 Here is a partial answer, which may point the way to a reasonable conjecture. Along the way I do answer one of your several questions. First a correction: the Ford domain is independent of$p$(up to covering transformation). It depends on the Fuchsian group alone. The Dirichlet domain$D(p)$depends, of course, on$p\$. Next some setup, designed to learn ...

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