The riemann-surfaces tag has no wiki summary.

**2**

votes

**1**answer

285 views

### Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...

**2**

votes

**2**answers

309 views

### Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...

**1**

vote

**2**answers

485 views

### Riemann surfaces with bounded curvature

Say there are metrics $g_n$ on a compact Riemann surface $\Sigma$ with bounded curvature and bounded area, or even with the same area element . What can we say about the 'limit' of $(\Sigma, g_n)$? ...

**5**

votes

**2**answers

546 views

### Riemann surfaces that are not of finite type

I am interested in studying Riemann surfaces that are not of finite type. By a non-finite type Riemann surface, I mean a Riemann surface that is not conformally equivalent to any Riemann sub-surface ...

**7**

votes

**3**answers

603 views

### Automorphisms of Riemann surface and mapping class

For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?

**-2**

votes

**1**answer

1k views

### Reducibility (or not) of algebraic curves [closed]

[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...

**0**

votes

**0**answers

215 views

### Is the absolute value of the j-invariant bounded from below on an annulus

Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip ...

**1**

vote

**1**answer

366 views

### Are Weierstrass points algebraic

Let $X$ be a compact connected Riemann surface of genus $g>0$.
Suppose that $X$ can be defined over a number field (as an algebraic curve). Then, is it clear that each Weierstrass point of $X$ is ...

**0**

votes

**2**answers

892 views

### Motivation behind defining the Ramification Divisor

I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function.
As I see the definition,
If $f$ is a meromrophic function between two ...

**0**

votes

**3**answers

902 views

### Classification compact Riemann Surfaces

I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?
In other words, is the complex structure ...

**2**

votes

**0**answers

210 views

### Is there an algebraic analogue of the degeneration of riemann surfaces in M_g

Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite alot. The idea is to ...

**1**

vote

**1**answer

335 views

### A theorem by Hopf on surfaces

I am reading a paper on Riemann surfaces and faced a problem about one of the refernces the author gave in the exlaination of one of the results.
Here is a summary of what I am reading:
Let $X_1$ ...

**2**

votes

**1**answer

427 views

### The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ ...

**2**

votes

**2**answers

261 views

### Reference request: parametrizing covers of the projective line

Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways.
For example, one could fix the number $r$ of branch points, the degree $n$ of the cover ...

**3**

votes

**2**answers

1k views

### Historical basis and mathematical significance of Riemann surfaces [closed]

It is written in Riemann Surfaces (Oxford Graduate Texts in Mathematics) by Simon Donaldson, that:
"[t]he theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination ...

**3**

votes

**4**answers

586 views

### $Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]

I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...

**0**

votes

**0**answers

256 views

### Complex structures on the upper half plane?

My question are related with this paragraph: (page 25 of A Fibre Bundle Description of Teichüller Theory by Earle and Eells.)
Let $X$ a Riemann surface of genus g>1, if $J\in M(X)$ ($M(X)$ space of ...

**1**

vote

**2**answers

244 views

### Any introduction to Teichmuller Space of T^2?

Is there any well written introduction for the modular space of complex structures on T^2?
Thank you!

**8**

votes

**1**answer

338 views

### Given a curve, under which condition is the set of gonal morphisms finite

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that ...

**7**

votes

**3**answers

970 views

### Hyperbolicity on Riemann Surfaces

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a ...

**2**

votes

**1**answer

444 views

### Branch points of a non-constant holomorphic map between compact riemann surfaces

While working on a project for mathematics I came across the following lemma: [Kock]
If $X$ is a curve defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow ...

**2**

votes

**1**answer

637 views

### A question on part of “An introduction to teichmuller spaces” by Imayoshi-Taniguchi

I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say:
Since
$\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued ...

**2**

votes

**0**answers

579 views

### Atiyah-Bott Yang-Mills connections

In Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces, a special case of what they do is to prove that Unitary Yang-Mills connections over a R.S $M$ are in bijective correspondence with ...

**4**

votes

**2**answers

646 views

### Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ?

Hi, my question is :
Let $D$ be the open unit disk in $\mathbb{C}=R^2$ and $f:D\to D$ be a real-analytic diffeomorphism. Let us think of the canonical embedding : $\mathbb{C}=R^2\subset \mathbb{C^2}. ...

**8**

votes

**1**answer

587 views

### Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\\!/SL(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,SU(2))/\\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, ...

**6**

votes

**2**answers

257 views

### Reference for openness of stable locus of holomorphic family of vector bundles on a compact riemann surface.

I am looking for a reference (or an easy explanation) for the openness of the stable locus of a holomorphic family of (holomorphic) vector bundles on a compact Riemann surface parametrized by a ...

**2**

votes

**1**answer

305 views

### Convergence radius of the q-expansion of the modular lambda function

Let $X(2)$ denote the compact Riemann surface obtained by compactifying $Y(2) = \Gamma(2)\mathfrak{h}$ by adding cusps.
The modular $\lambda$-function on the complex upper half plane $\mathfrak{h}$ ...

**9**

votes

**4**answers

1k views

### Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature.
For instance,
What is the spectrum of the ...

**1**

vote

**1**answer

332 views

### Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar

Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. ...

**12**

votes

**3**answers

1k views

### Is there an algebraic construction of the Quillen (determinant) Line Bundle?

Let's consider the moduli space of representations of $\pi=\pi_1(\Sigma)$ (a surface group) into $G$ (a lie group). Call this $X=\operatorname{Hom}(\pi,G)$, and let ...

**2**

votes

**1**answer

247 views

### Conformally immersed Riemann surfaces and foliations

I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 ...

**1**

vote

**1**answer

356 views

### Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states :
Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...

**0**

votes

**1**answer

458 views

### immersion: submanifold of complex manifold

Let $\alpha : \mathbb{C} \rightarrow M$ be an immersion and $M$ a $n$ dimensional complex manifold with complex structure $I$. Does then follow that $\alpha (\mathbb{C})$ is a one dimensional ...

**4**

votes

**1**answer

313 views

### Residual finiteness of fundamental groups of surfaces.

What is a simple way to prove that for any compact two-dimensional surface $S$ and an element $g$ in $\mathbb \pi_1(S)$ there exists a finite index normal subgroup $\Gamma\subset \pi_1(S)$ such that ...

**0**

votes

**1**answer

195 views

### Are transversely immersed PL surfaces Riemann surfaces?

I have a piecewise linear (PL) surface transversely immersed in $\mathbb{R}^3$; is this a Riemann surface in the sense that I can describe it with a local coordinate $z\in \mathbb{C}$? My basic ...

**5**

votes

**5**answers

2k views

### “Famous” 2d Riemannian manifolds with non-constant curvature

I'm looking for "famous" or otherwise well-known 2d Riemannian manifolds which have non-constant curvatures but have a non-trivial Killing vector field. Of course there are tons of spaces like these, ...

**5**

votes

**0**answers

462 views

### nth symmetric power of a Riemann surface and its Jacobian

Let $C$ be a Riemann surface of genus $g$, $Sym^{n}C$ be the nth symmetric power of $C$ with $n\geq 2g$, and $JC$ denote the Jacobian of $C$.
Question: Is it generally true that $Sym^{n}C\cong ...

**3**

votes

**6**answers

637 views

### Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...

**2**

votes

**0**answers

143 views

### Explicit construction of irreducible unitary connections

On a Riemann surface irreducible unitary connections on complex vector bundles of rank $n\geq 2$ are uniquley determined by their underlying holomorphic structure $\frac{1}{2}(\nabla+i*\nabla)$, where ...

**3**

votes

**1**answer

177 views

### number of ribbon structures (or punctured surfaces) on a graph

Suppose $G$ is connected undirected graph.
Does the calculation of the number of topologically distinct punctured surfaces that can arise from putting a ribbon structure on $G$ exist in the ...

**2**

votes

**3**answers

995 views

### Universal covers of punctured hyperbolic surfaces

Suppose S is a genus g surface with n punctures satisfying the hyperbolicity condition 2g + n - 2 > 0. If n > 0 the fundamental group of the surface is a free group on 2g + n - 1 := m generators.
If ...

**0**

votes

**1**answer

160 views

### Definition of k -quasisymmetric maps on S^1

I know the definition of k -quasi-symmetric maps $f$ on $R$,it is
there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$
So I just want to ...

**3**

votes

**5**answers

2k views

### Basic question about branch points on Riemann surfaces

If $X$ and $Y$ are Riemann surfaces (not necessarily compact), and $f:X\to Y$ is a holomorphic function, then it is obvious that the ramification points of $f$ in $X$ form a discrete subset of $X$. Is ...

**7**

votes

**1**answer

537 views

### Non-algebraic curve visualisation

Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert ...

**4**

votes

**1**answer

359 views

### Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps

If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get ...

**3**

votes

**4**answers

738 views

### Quotient Surface of A Hyperelliptic Involution

Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J > $ is a Riemann ...

**1**

vote

**1**answer

282 views

### Necessary condition for a branch point

If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend ...

**2**

votes

**0**answers

320 views

### Automorphisms of Riemann surfaces

It is well known that, the Hurwitz groups are quotients of the $(2,3,7)$ triangle group $\Gamma=\langle a,b\colon a^2=b^3=(ab)^7=1\rangle $ in $PSL(2,\mathbb{R})$. If $G$ is a Hurwitz group, then ...

**7**

votes

**3**answers

1k views

### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...

**3**

votes

**2**answers

666 views

### Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature

A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...