The riemann-surfaces tag has no usage guidance.

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### Compact surfaces with arbitrary gaps in spectrum

Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...

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**1**answer

71 views

### Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...

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164 views

### Finite orbits on an elliptic curve with two generic involutions

Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor.
Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an ...

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150 views

### Almost complex structures on compact surfaces

Let $M$ be a real manifold of dimension $n$ and $E\rightarrow M$ be a rank two vector bundle above it. Let $J_0$ and $J_1$ be two almost complex structures on $E.$ If there is a continuous ...

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676 views

### What are the possible automorphism groups of Riemann surfaces of low genus?

Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3?
I'm frustrated because there are papers that ...

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120 views

### Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces

By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet ...

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### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

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**1**answer

62 views

### Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...

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### Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = ...

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68 views

### $\eta$-invariants of Riemann Surface

I am curious about a concrete computation of $\eta$-invariants for Riemann surface, e.g. Torus.
Is there any nice review or notes talking about the computation? Or is it possible to express it as ...

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85 views

### Conformal welding of annuli

This question is similar to that stated in Conformal Welding Reference:
Let $\Sigma$ be a 1-dim compact connected complex manifold with boundary $\partial \Sigma= \partial\Sigma^+ \cup ...

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28 views

### Can we Foliate anti-de sitter space in 3 dimensions by riemann surfaces?

Can we foliate the anti-de sitter spacetime in 3 dimensions by the by the hyperbolic riemann surfaces, I think this is possible but stuck at finding the particular projective mapping that does this. ...

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**1**answer

52 views

### criterion for a differential of the third kind to be a logarithmic derivative of a function

Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential
of the third kind with ...

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181 views

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

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**1**answer

69 views

### Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$

Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem?
The only thing I can find is ...

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123 views

### Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...

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**1**answer

102 views

### Decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...

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84 views

### Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface.
Definition: A Quasiconformal surface $S$ is a ...

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224 views

### What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$.
Let $H$ be the Zariski closure of the ...

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### Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...

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**1**answer

388 views

### Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...

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130 views

### Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...

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131 views

### Translation surfaces & integer multiples of $\pi$

Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...

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### An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces

In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$:
$$
{\rm Tor}_{1,n}=\left\{
\big(\tau, ...

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**1**answer

339 views

### Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]

Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?

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205 views

### When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...

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111 views

### Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...

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### Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...

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169 views

### existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...

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### Analytic functions space on Riemann surface

I have some questions about the analytic function space on Riemann surface and distinguished varieties:
Let S be a compact Riemann surface and $\Omega\subset S$ be a domain with piecewise smooth ...

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215 views

### Ramification: Riemann surfaces vs Number fields

I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...

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### Projective curves of constant curvature

A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...

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### Kodaira-Spencer theory in d=1

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$.
To determine dimension of $T\mathcal M_g$,
start with a complex structure, which in some coordinates can be written
...

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118 views

### Dilatation of surface diffeomorphisms

Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism.
Define dilatation of $f_*$ as the largest ...

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### Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties:
For every two points in the plane there exists a unique geodesic joining them.
Every geodesic determines exactly two points on the ...

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391 views

### Is there a non-abelian version of the Torelli map?

Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M ...

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### Marten's proof of torelli theorem

I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 ...

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**1**answer

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### Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$

Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = ...

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422 views

### Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...

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vote

**1**answer

163 views

### How to understand a rooting of a dessin d'enfant?

As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...

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### Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...

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### Intersection of closed geodesics in hyperbolic surface

This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...

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### different proofs of the fact that compact riemann surface has a non-trivial meromorphic function

I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...

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### Connection between Strebel differentials, ribbon graphs, and Belyi maps

In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ...

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### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

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### Finding Riemannian metric explicitly from the conformal structure on a surface

Each Riemann surface structure $S'$ on a topological surface $S$ can be obtained from a Riemannian metric which looks like $ds^{2} = g_{11}dx^{2} + 2g_{12}dxdy + g_{22}dy^{2}$ in local coordinates ...

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### Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...

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### Riemann surface of $K_0(z)$

The question concerns the modified Bessel function of the second kind of order zero ($K_0(z)$).
How does its Riemann surface looks like? How can one evaluate its values on the "other" sheet(s)?

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### The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...

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222 views

### Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this:
Given divisors
$(\omega_1) = P_1 + 5 P_2 + 2 P_3,$
$(\omega_2) = 5 P_1 + P_2 + 2 P_3,$
...