The riemann-surfaces tag has no usage guidance.

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### Reference on Deligne-Mumford compactness for Riemann surfaces

I am working with closed degenerating hyperbolic Riemann surfaces, and I try to understand the compactification of the moduli space. Looking in different books, notably the one of Hummel, I now get a ...

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### Questions about dessin d'enfants, trees and their Shabat polynomials

This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume.
(Note: All of these ...

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### Request for some references exploring the connections of Riemann surfaces with medical imaging

I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical ...

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### Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a ...

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### Reference for hyperelliptic curves

I was reading a paper the other day that said that all automorphisms of a hyperelliptic curve are liftings of automorphisms of $\mathbb{P}^1$ operating on the set of branch points.
Can someone point ...

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### Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...

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### What prevents a cover to be Galois?

Let $f:X\rightarrow Y$ be a ramified cover of Riemann surfaces or algebraic curves over $\mathbb{C}$. My question is can one in terms of the ramification data of $f$, determine whether the cover is ...

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### Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...

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### Periods of translation surfaces

A translation surface is a Riemann surface equipped with a holomorphic 1-form $\omega$ and a Riemannian metric $g=\omega \bar \omega$ with conical singularities. It is well-known that there exists ...

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### The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...

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### What is a branched Riemann surface with cuts?

Edit: Let me restate the main claim being made in these two papers,
Consider the "branched" Riemann surface which has "n" sheets stuck along the intervals, $[z_i, z_{i+1}]$ for $i=1,..,2N$ then it ...

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### Cohomology of the genus 2 mapping class group

Is the cohomology of the genus 2 mapping class group (that is, the cohomology of the moduli stack $M_2$ of genus 2 curves) known? I'd be interested in references. The rational cohomology is known to ...

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### Approximation Runge's Theorem

Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...

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### Hyperbolic structures on once punctured tori

I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...

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### Octonions and the Fano plane.

Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?
http://upload.wikimedia.org/wikipedia/commons/2/2d/FanoPlane.svg
The symmetry group of the Fano plane is PSL(2,7), ...

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### A “Riemannian” analogue of Kobayashi metric?

Recall that Kobayashi metric is defined on any complex manifold $M$. This is a pseudo-metric according to which a tangent vector $v$ at $P$ has length at most $1$ if there is holomorphic map from ...

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### Non-compact Riemann surfaces and affine algebraic curves

Is every connected non-compact Riemann surface biholomorphically equivalent to an affine algebraic curve in some ${\mathbb C}^n$? I suspect that surfaces of infinite genus probably are not but could ...

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### Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...

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### Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be:
-Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function.
-Poincaré hyperbolic if it is covered by the unid disk.
Are this ...

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### Computing saddle connections in flat structures

Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ which start and end at a ...

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### Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$.
Is $X - \overline{D(x,r_x)}$ hyperbolic?

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### Complete curves in $M_g$ and Theta Characteristics

Let $g\geq 3$. Following the reference below, the locus of curves in $M_g$ with an effective even theta characteristic has codimension $1$. (Those are the curves $C$ with an effective line bundle $L$ ...

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### A question from Otto Forster's book on Riemann surfaces

I am reading section 14, A Finiteness Theorem of Otto Forster's book Lectures on Riemann Surfaces, and come across a problem on Theorem 14.15 on page 117. In the proof Forster introduces a function
...

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### Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces

I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, ...

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### The existence of meromorphic functions on Riemann surfaces

In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes:
It is a basic and highly nontrivial
result that a compact Riemann surface
has nonconstant meromorphic functions
on ...

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### On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...

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### Are morphisms of intersection graphs of circle packings harmonic?

Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...

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### Fuchsian groups and their normalizers

Let $\Gamma \leq PSL_2(\mathbb{R})$ be a Fuchsian group. What is the relation between $N(\Gamma) = \{ \alpha \in PSL_2(\mathbb{R}) \mid \alpha \Gamma \alpha^{-1} = \Gamma \}$ and $Aut(\Gamma ...

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### Injective morphism from an elliptic curve to $\mathbb CP^2$.

Let $E$ be the elliptic curve $x^3+y^3+z^3=0$.
Question. Are there injective morphisms $E\to \mathbb CP^2$ of arbitrary high degree?
Comments. 1) There are injective morphisms $E\to \mathbb CP^2$ ...

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### Fuchsian groups and automorphisms of Riemann surfaces

Let $\Gamma \subseteq PSL_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in ...

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### Injective morphism from curves to $\mathbb CP^2$

Is there a smooth compact complex curve that does not admit an injective holomorphic map to $\mathbb CP^2$ ? Let me stress, that the image of the curve in $\mathbb CP^2$ can have singularities.
I ...

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### When is a cyclic cover hyperelliptic?

Let us work over the complex numbers for simplicity. Consider a curve $C$ presented as a cyclic cover of some lower genus curve $C'$. When $C'$ has genus $0$, we can write $C$ as the normalization of ...

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### Reference for notation $H^0(C, mK)$

I am reading the draft of "Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups" and the author starts using the notation $H^0(C, mK)$ on page 3, without explaining it. As ...

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### Analytic curve on Riemann surface

Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...

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### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...

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### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

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### representation of teichmuller space Teichmuller space

I want to study representation of teichmuller space of surface of genus g in psl(2,R).
can you suggest any good references.

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### Complex curves covered by smooth plane curves

Question: Is it true that for every smooth compact complex curve $C$ there exists a smooth curve $C'$ in $\mathbb CP^2$ that admits a non-trivial morphism (i.e. holomorphic map) $C'\to C$?
...

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### Automorphisms of higher-genus Riemann surfaces act nontrivially on homology (Reference Request)

The following is a folklore result : Let $X$ be a compact Riemann surface of genus at least $2$ and let $f : X \rightarrow X$ be a biholomorphism. Then $f$ acts nontrivially on $H_1(X;\mathbb{Z})$.
...

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### Vortex equations on cylinder

Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In ...

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### Is the set of all smoothed closed simple curves on $\mathbb{R}^2$ a manifold?

In the studies of active contours they describe the set of all simple smooth closed curves on $\mathbb{R}^2$ to be a Riemannian Manifold $M$. The tangent space at a curve $c$, $T_cM$ is a set of ...

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### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...

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### growth of energy of eigenfunctions on hyperbolic surface

I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface.
Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and ...

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### How does a moduli interpretation give an analytic object an algebraic structure?

I remember hearing this in other contexts, but I encountered it again when reading Elkies' paper "Shimura Curve Computations", where on page 10, he says that:
"We now return to the Shimura curves ...

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### Rational curved lying in the boundary of Deligne-Mumford compactification $\bar M_g$

Let $\bar M_g$ be the Deligne-Mumford compactifiction of the moduli space of complex genus $g$ curves $M_g$. Is this correct that through every point of the boundary $\bar M_g\setminus M_g$ passes a ...

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### Regular (or complex analytic) functions on M_3

Let $M_3$ be the moduli space of genus three curves over $\mathbb C$.
Are there non-constant regular functions of this space? What about complex analytic functions?
This question is prompted by the ...

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### How do you find the genus of a Fuchsian group derived from a quaternion algebra?

Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$
Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form:
$$\langle ...

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### Does every smoothly embedded surface $\mathbb{R}^3$ inherit a natural complex structure, and if so, which one?

Smoothly embed a genus g surface in $\mathbb{R}^3$, and pick a normal vector pointing "out" of the surface at each point. Then on each tangent plane, I have a map which rotates the tangent plane 90 ...

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### Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?
I know that in this case the universal cover is the hyperbolic plane ...

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### Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

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 : ...