Questions tagged [riemann-surfaces]

Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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Cutting and pasting in Galois theory

I want to ask who was the first to use cut-paste construction in Galois theory. This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois ...
Lior Bary-Soroker's user avatar
1 vote
2 answers
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Hopf Tori in $S^3$

By means of the Hopf fibration $\pi: S^3 \rightarrow S^2$, U. Pinkall showed that every compact Riemannian surface of genus one can be conformally embedded in $S^3$. More precisely: Let $p$ be a ...
user18589's user avatar
11 votes
4 answers
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Growth of smallest closed geodesic in congruence subgroups?

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$. How does the lower bound for the length of primitive geodesics on $\...
Marc Palm's user avatar
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Is the cardinality of occuring torsion subgroups in cofinite lattices in SL(2,R) bounded?

Let $\Gamma$ be a cofinite lattice in $PSL(2,\mathbb{R})$ with torsion subgroup $H$. Is the a uniform bound on the cardinality of $H$?
Marc Palm's user avatar
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3 votes
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Questions on a Certain Branched Cover of the Two-sphere

I have the following questions: Suppose the compact Riemann surface $C$ is an n-fold branched cover of $\mathbb{P}^1$ branched at exactly four points $x_1,x_2,x_3$ and $x_4$. I believe that $C$ is a ...
D. S. Park's user avatar
6 votes
3 answers
651 views

Questions on 3-manifolds with a given boundary

I have the following question: For a given two-dimensional Riemann surface $C$, Is there a way to classify all topologically distinct three-dimensional compact manifolds $M$ whose boundary is $C$, i....
D. S. Park's user avatar
2 votes
1 answer
361 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 $g$ ...
Ariyan Javanpeykar's user avatar
2 votes
2 answers
606 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 ...
Jaikrishnan's user avatar
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3 votes
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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 $(\Sigma, g_n)$? ...
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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 ...
Jaikrishnan's user avatar
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8 votes
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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?
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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 ...
Anirbit's user avatar
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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 $$\{...
Taicho's user avatar
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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 ...
Taicho's user avatar
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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 ...
Anirbit's user avatar
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4 answers
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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 ...
Abramo's user avatar
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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 ...
Tamed's user avatar
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1 answer
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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 $X_1$ ...
yaa09d's user avatar
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3 votes
1 answer
852 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 $X$ ...
Ariyan Javanpeykar's user avatar
2 votes
2 answers
507 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 ...
Ariyan Javanpeykar's user avatar
4 votes
2 answers
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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 ...
Sadiq Ahmed's user avatar
3 votes
4 answers
1k 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 ...
Anirbit's user avatar
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Any introduction to Teichmuller Space of $T^2$?

Is there any well written introduction for the modular space of complex structures on $T^2$?
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1 answer
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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 ...
Ariyan Javanpeykar's user avatar
10 votes
3 answers
2k 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 ...
ght's user avatar
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1 vote
1 answer
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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 \...
Konrad Pilch's user avatar
3 votes
1 answer
780 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 ...
daniel's user avatar
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0 answers
776 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 ...
Vamsi's user avatar
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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}. ...
Analysis Now's user avatar
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14 votes
1 answer
874 views

Is there an algebraic description of Yang-Mills measure?

Consider $X=\operatorname{Hom}(\pi,\mathrm{SL}(2,\mathbb C))/\! \!/\mathrm{SL}(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,\mathrm{SU}(2))/\mathrm{SU}(2)$, where $\pi$ is a surface group. Note that ...
John Pardon's user avatar
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6 votes
1 answer
367 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 (...
bavajee's user avatar
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2 votes
1 answer
536 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 $\mathfrak{h}$ ...
Ariyan Javanpeykar's user avatar
11 votes
4 answers
3k 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 ...
ght's user avatar
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2 votes
1 answer
563 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. (...
Ariyan Javanpeykar's user avatar
17 votes
3 answers
3k 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 $Y=\operatorname{Hom}(\pi,G)/\\!/G$...
John Pardon's user avatar
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4 votes
1 answer
342 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 ...
cduston's user avatar
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1 answer
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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 $d_X,d_Y$ ...
Analysis Now's user avatar
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0 votes
1 answer
896 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 ...
gregor's user avatar
  • 23
4 votes
1 answer
522 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 $...
aglearner's user avatar
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0 votes
1 answer
245 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 ...
cduston's user avatar
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7 votes
5 answers
3k 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, ...
Daniel's user avatar
  • 362
5 votes
0 answers
726 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 JC\...
YOURS's user avatar
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4 votes
6 answers
902 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) $ ...
Analysis Now's user avatar
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1 vote
0 answers
168 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 ...
Sebastian's user avatar
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3 votes
1 answer
209 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 ...
Ivan's user avatar
  • 47
6 votes
3 answers
2k 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 ...
alethea's user avatar
  • 61
0 votes
1 answer
194 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 ...
Analysis Now's user avatar
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6 votes
6 answers
4k 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 ...
rfauffar's user avatar
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7 votes
1 answer
786 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 ...
Sasha's user avatar
  • 1,343
5 votes
1 answer
702 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 ...
Analysis Now's user avatar
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