# 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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### Fenchel–Nielsen coordinates vs Fock–Goncharov coordinates

Consider an orientable surface $S$ and its Teichmüller space $S$, which is the space of representations of its fundamental group $T(S)=\{\rho: \pi_1(S) \to \operatorname{SL}(2,\mathbb{R})\}$. Fock and ...

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### Curves having only one linear system realizing its gonality

$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree ...

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### Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....

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### Action of the mapping class group on curves and triangulations

Consider an orientable surface $S$ of arbitrary genus, possibly with boundaries, and with marked points and/or punctures. I will assume that every boundary has at least one marked point so that the ...

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### Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees

I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \...

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### Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...

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### Inner product on global sections of positive line bundle

Let $\Sigma = S^2$ be thought of as a Riemann surface, and let $L$ be a Hermitian line bundle on $\Sigma$ with curvature $2$-form $-2 \pi i \Omega \in \Omega^2(\Sigma, \mathbb{R})$. Then $L$ is a ...

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### Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...

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### When is a compact orbifold Riemann surface a global quotient of a Riemann surface

While reading the paper Seifert Fibred Homology 3-Spheres and the Yang-Mills Equations on Riemann Surfaces with Marked points by M. Furuta and B. Steer, I stumbled upon the following statement:
Any ...

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### Fenchel-Nielsen length-length coordinates on Teichmueller space?

Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...

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### Triangulations with discrete metrics and conformal equivalence

A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...

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### definition of generic function

what is definition of generic function in following paper ? i need a reference for definition generic function .
"A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...

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### Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]

$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are ...

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### Do surface groups embed into PSL_2 over a real quadratic integer ring?

$\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be ...

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### operations on matrices preserving the property of being the Riemann matrix of a surface

I have heard about the Schottky problem and the related Novikov's conjecture about the characterization of matrices in the Siegel upper half-space which are indeed the Riemann matrix of a compact ...

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### Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...

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### Can we always find coordinates on a surface such that $K=K(u-v)$?

Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function ...

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### Graphs from the point of view of Riemann surfaces

I was listening to the lecture "Graphs from the point of view of
Riemann surfaces" by Prof. Alexander Mednykh. I am looking for references for the basics of this topic. Any kind of ...

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### Nonrepresentability by radicals and entire (or meromorphic) functions of algebraic functions

It is known that an algebraic function with non-solvable monodromy group can not be represented by radicals. Where can we find a detailed proof about the nonrepresentability by radicals and entire (or ...

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### The real part of the period of an elliptic curve

Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \...

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### Simplest Liouville Manifold not of Finite Type, or - Liouville Cobordism Structure on Pair of Pants?

I've been trying to produce the simplest possible example of a Liouville manifold which wouldn't be of finite type (a Liouville manifold is said to be of finite type if its skeleton is compact), and ...

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### A Riemann surface is automatically paracompact

[A question I remember from many years ago.]
Definition
A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...

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### Schwarzian derivative, accessory parameters, projective connections

I am looking at the following Riemann surface (let's call it $M$),
\begin{equation}
y^n=\frac{(x-x_1)(x-x_3)}{(x-x_2)(x-x_4)}
\end{equation}
which is a Riemann surface of genus $n-1$. It can be ...

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### Is there a decision procedure for analytic continuation?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of ...

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### Selberg zeta function analytic expressions

Consider the following algebraic equation,
$$
y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}
$$
which is a Riemann surface of genus $n-1$ (after compactifying). The classical retrosection theorem due to ...

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### Reverse residue theorem without using Serre's duality

In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text):
Let $\{a_1, \dots,a_n\}$ be a set of points in ...

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### Existence of continuous family of uniformising parameters

I asked this question on MSE a while ago but didn't receive any useful answers.
Suppose I have a $1$-parameter family continuous maps $f_t: \mathbb{S}^2\rightarrow \mathbb{C}P^1$ from a topological $2$...

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### Two definitions of Teichmüller space: relative isotopy or not?

The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism.
The definition on ...

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### "Pushforward" of a universal curve by a map (Riemann surfaces)?

For a Riemann surface $X$ with a Beltrami form $\mu \in M(X),$ ($M(X)$ is the space of Beltrami forms on the Riemann surface), we can define the Riemann surface $X_\mu.$ In John H. Hubbard's ...

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### Products of eigenfunctions on compact Riemann surfaces

Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...

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### Uniformization of triangulation on a sphere up to Moebius transformations

This is not the most precise question but rather a hope that someone has seen something like this.
I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...

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### Laplace-Beltrami of the mean curvature

For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...

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### Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

I am trying to visualize the genus-two Riemann surface given by the curve
$$
y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}.
$$
We can regard this surface as a three-fold cover of the sphere with four ...

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### Explicit example de Rham moduli space of connections

Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:
-$M_{Dol}$ the moduli space of stable ...

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### $L^\infty$ norm of Schwarzian derivative implies univalence?

On page 26 of these notes (Riemann surfaces, dynamics and
geometry) there is a theorem 2.13 which says
Let $f: \mathbb H \to \hat{\mathbb C}$ be a holomorphic map, satisfying $\|Sf\|_{\infty} <1/2$...

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### Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

According to Riemann surfaces, dynamics and
geometry
by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by
$$
\|\phi\|_p = \left(\...

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### Beurling’s extremality criterion for curves: two versions

I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below,
$$
\ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|.
$$
"Extremal&...

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### Recovering a family of rational functions from branch points

Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:X\rightarrow Y$ of degree $d$ with branch points ...

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### Relationship between Dolbeault and de Rham cohomology on Riemann surface

A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...

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### Dessins d'enfant of Dynkin diagrams?

Dessins d'enfant have a nice particular case of Shabat trees, where we take a tree, bicolor it, and get a polynomial map.
A very famous set of trees are the Dynkin diagrams. I wonder what are the ...

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### Triangulating Riemann surfaces by using non-constant meromorphic functions

Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...

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### Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces.
Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...

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### A basis of holomorphic differentials on Fermat curves

I am currently reading the paper "Holomorphic Differentials of Generalized Fermat Curves" by Rubén Hidalgo. The case I am interested in is that of a classical Fermat curve $F_k$, which in ...

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### To integrate elliptic integral, we glue two Riemann surface to make torus

To deal with elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this in order to avoid indeterminacy of integral, in other word, to avoid the condition ...

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### Composition of coproduct and product in Lagrangian Floer (co)homology

Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...

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### Finitely connected orientable surface

Let $(M,g)$ be a finitely connected orientable complete Riemannian surface, that is, $M$ is homeomorphic to a compact orientable surface $\Sigma$ minus $k \geq 1$ points. Do you have references or a ...

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### Self homeomorphism of $\mathbb CP^1$ holomorphic a.e

Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism holomorphic on a connected open subset $U\subset \mathbb CP^1$ with $\mathbb CP^1\setminus U$ of zero measure.
Is it true that $\...

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### Universal cover of finetely connected surface with boundary

Let $M$ be a finetely connected orientable surface with compact boundary. This means $M$ is homeomorphic to a compact orientable surface $\Sigma$ of genus $g \geq 0$ minus $r \geq 1$ points and minus $...

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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 ...