The riemann-surfaces tag has no wiki summary.

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### 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 ...

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**1**answer

322 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 ...

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**4**answers

650 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 ...

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**1**answer

259 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 ...

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305 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 ...

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### 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 ...

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572 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 ...

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**1**answer

339 views

### Estimating laplace-beltrami spectra for a graph surface in $R^3$

Consider a surface $\Gamma$ in $R^3$. The surface $\Gamma$ is a graph, i.e. $\Gamma = (x,y, h(x,y))$, for $x \in R^2$ and some smooth function $h$, where $h$ and all its derivatives are periodic on ...

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### Automorphisms of Riemann Surfaces

Izumi Kuribayashi and Akikazu Kuribayashi have classified all groups of automorphisms of compact Riemann surfaces of genus 3,4,5. (J. Pure Applied Alg.65(3)-Sept.1990, and J. Alg.134(1) Oct.1990)
...

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**1**answer

227 views

### Equivalence of Branched Coverings

For equivalence of unbranched coverings of topological spaces, there is a criteria:
Two coverings (unbranched) $p_1\colon Y_1\rightarrow X$ and $p_2\colon Y_2\rightarrow X$ are equivalent iff for ...

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**0**answers

258 views

### Calculation of dimension of holomorphic quadratic differentials as in Gardiners book

In Frederick Gardiner's book Teichmuller Theory and Quadratic Differentials, P.27-28, Chapter 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n $ ( by using Riemann-Roch theorem ). Now for open annulus ...

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**3**answers

686 views

### Groups acting on Riemann Surfaces

By Hurwitz theorem, order of a group $G$ of automorphisms (conformal homeomorphisms) of a compact Riemann surface of genus $g\geq 2$ is bounded above by $84(g-1)$.
1. Is there any example of a ...

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**1**answer

605 views

### Basic Questions about Teichmuller's theorem/quadratic differentials

I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...

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**2**answers

179 views

### The antipodal action on a connected one dimensional manifold

When I am reading one paper, I have met the following statement:
It is impossible to define a $Z_{2}\times Z_{2}$ action on a connected closed curve on a compact Riemann surface.
The claim is ...

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**3**answers

6k views

### Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...

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519 views

### Conformal Welding Reference

I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique ...

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**2**answers

385 views

### Action of a group on a Riemann surface

If $G$ is a finite group of order $n$, and acting on a compact Riemann surface $X_{g'}$ ($g'$ is genus), then $X_{g'}/G$ is another compact Riemann surface, $X_g$, of genus $g$.
Also then $X_{g'} ...

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2k views

### Problem in Rick Miranda: finding genus of a Projective curve

I asked the following question in stack exchange (http://math.stackexchange.com/questions/21164/problem-in-rick-miranda-finding-genus-of-a-projective-curve) a few days ago, but didnt get any solution. ...

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**2**answers

645 views

### A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

Hello,
I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...

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860 views

### some questions on Riemann surface

There are several puzzling questions on Riemann surface for me: Q.1 Definition of Riemann surface can be given in at least two ways: Def.1) it is a complex one dimensional manifold; Def.2) for each ...

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**1**answer

226 views

### Can we prove $ Aut(S_g) , g \geq 2 $ is finite in the following way ?

I was trying to prove that $ Aut( S_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way :
For fixed $M $ ( positive ) there are finitely many , say $ k $ number of ...

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**0**answers

140 views

### What is the parametrix for the d-bar operator on Sobolev spaces?

I am attempting to review a certain proof of the Riemann-Roch theorem, a key step of which is "elliptic bootstrapping." In particular, let $X$ be a compact Riemann surface, and $E$ be a holomorphic ...

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### Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...

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**1**answer

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### Which Riemann surfaces arise from the Riemann existence theorem?

The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...

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**1**answer

437 views

### Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello ,
I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I ...

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**1**answer

556 views

### Riemann's theorem on theta

Is there a low-tech way to compute the chern class of the theta bundle, i.e. the bundle associated to theta divisor? Here, the theta divisor is the analytic set of holomorphic line-bundles in ...

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**2**answers

286 views

### Why a non-simple geodesic in a Y-piece is NOT homotopic to a common perpendicular to the geodesic boundaries ?

This is a basic question, still I dare to ask :
Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want ...

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**1**answer

289 views

### Nielsen extension of Riemann surface

Let $S$ be a riemann surface.
If S has idea boundary curves,then the intrinsic metric on $S$ can be defined by the restriction to $S$ of poincare metric of the double of $S$.
Also this metric can be ...

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**1**answer

244 views

### Figure eight geodesic on a pair of pants/Y-piece

Consider a figure-eight geodesic $\delta $( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $ \gamma_i$ and three perpendiculars between ...

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**2**answers

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### Normalisations of singular plane algebraic curves

In what follows assume that the base field is $\mathbb{C}$
Background: Let $C$ be an irreducible plane algebraic curve, $S$ the set of singular points. There exists a Riemann surface $\tilde{C}$ and ...

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### Teichmuller modular forms and number theory

Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?

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### Bounding the modular discriminant of an elliptic curve in the j-invariant

Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert ...

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### finite-dimensionality of cohomology groups on compact riemann surfaces

does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...

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### Translation surfaces

I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete ...

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### An analytic proof of the De Franchis theorem

The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...

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476 views

### Poles of Kloosterman Zeta Function

I am a string theorist who has encountered the following number theory problem in my
research.
Consider the sums
$$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)$$
and
$$Z_-(s) = \sum_{p=1}^\infty ...

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608 views

### A question in R.C.Penner's paper about Teichmuller space

In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ labelled boundary ...

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309 views

### Books about the spectra of non-compact Riemann surfaces

Hello,
Thanks for reading my question ! Could anybody give me some references ( books, papers containing elementary results etc ) on the eigen values and eigenspectra of NON-compact Riemann surfaces. ...

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**1**answer

759 views

### Question related to the moduli space of Riemann surfaces and a fibration

If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1_{g} ...

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**2**answers

486 views

### Geodesics on zero-curvature regions of closed surfaces of genus > 1 of non-positive curvature

Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover ...

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648 views

### Compact cover of a Hausdorff compact space

In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact ...

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**1**answer

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### Interpolation on real Riemann surfaces

Background:
Generalizing the notion of upper half plane to compact Riemann surfaces:
Suppose $p(x,y) \in \mathbb{R}[x,y]$ is a polynomial in 2 variables with real coefficients, defining a smooth ...

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**1**answer

586 views

### Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except ...

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### Moduli space of semistable bundles

It is well-known that the space of S-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface M is $CP^3$ (more concretely ...

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### Representation of the group of automorphisms on the holomorphic forms

Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and ...

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### Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not ...

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423 views

### Flat regions on surfaces of genus greater than 1

Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...

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**1**answer

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### How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?

How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...

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### Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...

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725 views

### To differently gluing of two Riemann surfaces with boundary we get different surfaces

If $M,N$ are two Riemann surfaces with boundary, then we can glue them along one of each of their boundary component, which is $S$, to form a new Riemann surface with boundary, but for different ...