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.
677
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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 ...
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2
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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 ...
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4
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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 $\...
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1
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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 $H$?
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2
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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 ...
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3
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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....
2
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1
answer
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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 $g$ ...
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2
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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 ...
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2
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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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2
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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 ...
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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 ...
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0
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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 $$\{...
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1
answer
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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 ...
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2
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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 ...
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4
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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 ...
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0
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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 ...
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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 $X_1$ ...
3
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1
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852
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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
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2
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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 ...
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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 ...
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4
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$Aut(\mathbb{CP}^n)$ [..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 ...
2
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2
answers
372
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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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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 ...
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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 ...
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1
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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 \...
3
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1
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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 ...
4
votes
0
answers
776
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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 ...
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2
answers
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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}. ...
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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 ...
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1
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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
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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 $\mathfrak{h}$ ...
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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 ...
2
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1
answer
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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. (...
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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 $Y=\operatorname{Hom}(\pi,G)/\\!/G$...
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1
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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 ...
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1
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557
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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 $d_X,d_Y$ ...
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1
answer
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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 ...
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1
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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
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1
answer
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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 ...
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"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, ...
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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\...
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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) $ ...
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0
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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 ...
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1
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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 ...
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3
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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 ...
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1
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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 ...
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6
answers
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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
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1
answer
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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 ...
5
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1
answer
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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 ...