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2
votes
2answers
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
2answers
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
4answers
571 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
0answers
252 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
2answers
238 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
1answer
324 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
3answers
947 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
1answer
425 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
1answer
630 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
0answers
570 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
2answers
625 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
1answer
572 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
2answers
252 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
1answer
295 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
4answers
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
1answer
320 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
3answers
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
1answer
246 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
1answer
344 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
1answer
416 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
1answer
311 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
1answer
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
5answers
1k 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
0answers
452 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
6answers
607 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
0answers
142 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
1answer
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 ...
1
vote
3answers
853 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
1answer
158 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
5answers
1k 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
1answer
525 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 ...
3
votes
1answer
342 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 ...
2
votes
4answers
703 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
1answer
275 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
0answers
311 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 ...
6
votes
3answers
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
2answers
631 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 ...
3
votes
1answer
356 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 ...
12
votes
2answers
1k views

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) ...
4
votes
1answer
230 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 ...
1
vote
0answers
300 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 ...
1
vote
3answers
704 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 ...
3
votes
1answer
644 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 ...
1
vote
2answers
182 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 ...
25
votes
3answers
7k 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 ...
11
votes
3answers
542 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 ...
2
votes
2answers
397 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'} ...
17
votes
6answers
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. ...
0
votes
2answers
665 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, ...
3
votes
2answers
894 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 ...