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I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete …

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 …

Hello, Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2- …

Hello, Thanks for reading my question ! Could anybody give me some references ( books, papers containing elementary results etc ) on the eigen values and eigenspectra of NON-compact Riemann surfaces. …

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 …

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 …

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) $ …

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 $ …

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}. …

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 …

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 …

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 …

This is a basic question, still I dare to ask : Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want …

I was trying to prove that $ Aut( S_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way : For fixed $M $ ( positive ) there are finitely many , say $ k $ number of …

Consider a figure-eight geodesic $\delta $( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $ \gamma_i$ and three perpendiculars between th …