# Questions tagged [riemann-surfaces]

Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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### Question in the proof of Hilbert's theorem

I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the ...
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### Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall ...
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### Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, ...
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### Visualizing holomorphic differentials on a compact Riemann surface?

It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...
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### Distance Metric on a Polytope

Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...
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### References on Hyperbolic Geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
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### References on Riemann surfaces

I have asked the question in MSE, but did not get an answer. I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
192 views

### Branched covers of the sphere branched over few points

Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...
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### Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)

In Riemann's publication about Abelian functions 'Theorie der Abelschen Functionen' (Here the original paper in german) at the end of Chapter 4, part 2 is clamed that for every Riemann surface $T$ and ...
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### Meromorphic functions on a modular curves of genus $0$ that take each value exactly once

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$, and let $\mathfrak H$ be the upper half-plane. Let $X(\Gamma)$ be the compactification of $\Gamma\backslash\mathfrak H$. Then ...
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### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it. Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...
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### Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
Does the following statement hold: Statement: For any $\epsilon > 0$, there exist a number field $k$ of degree $d_{\epsilon}$ over $\mathbb{Q}$ and an arithmetic hyperbolic surface  $\Gamma$ ...
Define the set $\Omega := (-\epsilon,\epsilon) \times (-1,1)^{n-1}$, and define $\Gamma := \{-\epsilon,\epsilon\} \times (-1,1)^{n-1} \subset \partial \Omega$. Suppose I have two functions \$u,v \in C^...