The riemann-surfaces tag has no wiki summary.

**-1**

votes

**0**answers

42 views

### Problems concerning meromorphic 1 form on Riemann surface [on hold]

1. For a compact Riemann surface $X$, let the genus of $X$ be the dimension of the space of holomorphic one forms on $X$. How to compute the genus of the Riemann sphere and a complex torus.
2. ...

**23**

votes

**5**answers

2k views

### 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 ...

**5**

votes

**1**answer

368 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 ...

**1**

vote

**0**answers

105 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 ...

**5**

votes

**1**answer

108 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 ...

**1**

vote

**0**answers

47 views

### 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, ...

**1**

vote

**1**answer

269 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?

**3**

votes

**1**answer

173 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 $P$ ...

**0**

votes

**0**answers

107 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 ...

**11**

votes

**0**answers

112 views

### 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 ...

**0**

votes

**1**answer

159 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 ...

**0**

votes

**0**answers

55 views

### 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 ...

**1**

vote

**0**answers

194 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 ...

**4**

votes

**2**answers

187 views

### 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 ...

**0**

votes

**0**answers

108 views

### 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
...

**2**

votes

**2**answers

113 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 ...

**6**

votes

**2**answers

190 views

### 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 ...

**8**

votes

**4**answers

364 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 ...

**1**

vote

**0**answers

140 views

### 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 ...

**3**

votes

**1**answer

151 views

### 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) = ...

**9**

votes

**3**answers

402 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 $\mathbb{H}/[G,G]$?
...

**1**

vote

**1**answer

153 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 ...

**0**

votes

**0**answers

190 views

### 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 ...

**2**

votes

**1**answer

116 views

### 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 ...

**3**

votes

**0**answers

142 views

### 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 ...

**3**

votes

**1**answer

116 views

### 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 ...

**0**

votes

**0**answers

209 views

### 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 ...

**1**

vote

**0**answers

96 views

### 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 ...

**1**

vote

**1**answer

113 views

### 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 ...

**0**

votes

**0**answers

50 views

### 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)?

**1**

vote

**0**answers

112 views

### 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 ...

**1**

vote

**2**answers

218 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
$(\omega_1) = P_1 + 5 P_2 + 2 P_3,$
$(\omega_2) = 5 P_1 + P_2 + 2 P_3,$
...

**4**

votes

**1**answer

150 views

### Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...

**3**

votes

**0**answers

224 views

### Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...

**17**

votes

**2**answers

825 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...

**1**

vote

**3**answers

297 views

### A special case of the uniformization theorem

I am interested in a proof of the following fact :
Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is conformally equivalent to the Riemann sphere.
Of course, this ...

**0**

votes

**1**answer

125 views

### Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri.
I quote from the paper-
Can someone please explain how does any non-zero homomorphism ...

**1**

vote

**2**answers

266 views

### Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$

in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but ...

**4**

votes

**1**answer

300 views

### searching for an elementary proof a complex analysis result

Given a function $ g $ entire on the whole complex plane $ C $, it is possible to find an entire function $f $ such that $ f(z+1) -f(z)=g(z) $. The proof can be given using riemann ...

**3**

votes

**1**answer

143 views

### Does every canonical decomposition of the intersection form come from a canonical homology basis?

Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...

**1**

vote

**0**answers

64 views

### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...

**1**

vote

**1**answer

133 views

### Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...

**0**

votes

**0**answers

76 views

### Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form:
Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...

**3**

votes

**3**answers

181 views

### Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?

I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...

**1**

vote

**2**answers

672 views

### Spicing up Riemann surfaces course (revised)

I am a master's student planning to write a master's thesis on Riemann surfaces. I plan to study Forster's Lectures on Riemann surfaces. What side topics could one study to spice up the thesis? I am ...

**1**

vote

**1**answer

79 views

### Generators for the affine automorphism group of the octagon

Consider an octagon $O$ with opposite edges identified. Lemma 3.2.4 of (1) (subscription link) claims that the affine automorphism group of $O$ is generated by $D_8$ and the shear $\sigma$ such that, ...

**0**

votes

**1**answer

109 views

### A question for hyperbolic metric in the proof for Bohr's lemma

Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question:
Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on ...

**2**

votes

**2**answers

280 views

### Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two.
Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve.
Is $Y$ hyperelliptic?
More ...

**2**

votes

**1**answer

119 views

### Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface

I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows.
Suppose $X$ is a Riemann surface that admits a green's function (i.e. ...

**3**

votes

**0**answers

113 views

### Criterion for a subgroup of $PSL_2(\mathbb R)$ to be Fuchsian

Let $\Gamma$ be a finitely generated subgroup of $PSL_2(\mathbb R)$.
I'm looking for effective criteria (idealy necessary and sufficient but just necessary would be a good start) ensuring that ...