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2
votes
1answer
293 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
312 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
245 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
337 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
406 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
305 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
194 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
422 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
595 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
141 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
173 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
805 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
517 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
329 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
679 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
270 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
309 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
609 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
350 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
285 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
697 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
626 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
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 ...
25
votes
3answers
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 ...
11
votes
3answers
532 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
393 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
656 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
878 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 ...
2
votes
1answer
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 ...
1
vote
0answers
144 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 ...
7
votes
1answer
1k views

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 ...
6
votes
1answer
1k views

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 $\phi: \pi_1(X \backslash \Delta) \to S_d$ ...
1
vote
1answer
442 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 ...
3
votes
1answer
569 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 ...
2
votes
2answers
289 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 ...
0
votes
1answer
295 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 ...
0
votes
1answer
246 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 ...
1
vote
2answers
960 views

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 ...
12
votes
3answers
2k views

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?
6
votes
2answers
739 views

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 ...
3
votes
2answers
485 views

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 ...