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
questions
8
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2
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Holomorphic maps from a Riemann surface of infinite genus
Let $X$ be a Riemann surface of infinite genus and let $n$ be an arbitrarily large natural number.
Do there always exist a closed Riemann surface $Y$ of genus greater than $n$ and a nonconstant ...
3
votes
1
answer
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Existence of covering isomorphism
Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...
7
votes
1
answer
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Riemann uniformization theorem (limit case)
Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus ...
-1
votes
1
answer
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Related to the Schwarz Christoffel map
With the help of the Schwarz-Christoffel map, for a given polygon (given angle), we can find some points on the boundary of the upper half plane, such that a particular Schwarz-Christoffel map takes ...
1
vote
0
answers
69
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Diffeomorphism induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...
2
votes
1
answer
104
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Pair of laminations that fill on a closed surface
Let $S$ be a hyperbolic surface of genus $g \geq 2$.
A discrete geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics.
Let $L_{1}$ and $L_{2}$ be two discrete geodesic laminations ...
6
votes
1
answer
473
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Topology change induced by small perturbation
Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...
2
votes
0
answers
121
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Is there an extension of Ogg's results to surfaces of Genus 1
The first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is ...
2
votes
1
answer
143
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Does moving a small enough distance in Teichmüller space change the marking?
Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \...
3
votes
1
answer
246
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Meromorphic function on the Riemann surfaces
Let $V$ be a Riemann surface, $x\in V$, and $B:=B(x,r)$ some small ball (in a local chart). It is well known that there is a meromorphic function $f$ on $V$ with the only pole at $x$. What I’d like to ...
3
votes
1
answer
401
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Relationship between two kinds of classifications of Riemann surfaces
There are two kinds of classifications of Riemann surfaces.
Classification 1: Let $M$ be a Riemann surface. We will call $M$:
elliptic iff $M$ is compact (= closed);
parabolic iff $M$ is not compact ...
1
vote
2
answers
143
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Showing (branched complex) affine surfaces admit complex affine structures
I am reading the article of Apisa-Bainbridge-Wang on moduli spaces of complex affine structures.
In Definition 2.5 on page 7, they define a (branched complex) affine surface to be a tuple $(X, P, \chi,...
8
votes
1
answer
355
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Moduli of flat connections vs Teichmuller space on surfaces
The dimension of moduli of flat $SU(2)$-connection is the same as the dimension of Teichmuller space, both on a surface of genus $g$; namely both dimensions are equal to $6g-6$. Is this a coincidence ...
4
votes
0
answers
255
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Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
3
votes
1
answer
167
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Identifying the conformal equivalence class of a 2-torus subgroup of the cubical 3-torus
Let K, L, M be integers with gcd(K,L,M) = 1. They determine a connected Lie subgroup G = G(K,L,M) of the cubical 3-torus (ℝ/ℤ)3 via
G = {(x,y,z) ∊ (ℝ/ℤ)3 | Kx + Ly + Mz = 0}
(where 0 denotes the ...
4
votes
0
answers
143
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When is maximal analytic continuation a Zariski open set?
I have following question in mind. I apologize if this question is too obvious or naive.
Let $U$ be an Euclidian open set of $\mathbb{C}^*$ and $f:U\to U \times F\subset\mathbb{C}^4$ be an analytic ...
3
votes
1
answer
218
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Fenchel–Nielsen coordinates vs Fock–Goncharov coordinates
Consider an orientable surface $S$ and its Teichmüller space $S$, which is the space of representations of its fundamental group $T(S)=\{\rho: \pi_1(S) \to \operatorname{SL}(2,\mathbb{R})\}$. Fock and ...
2
votes
1
answer
232
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Curves having only one linear system realizing its gonality
$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree ...
11
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3
answers
726
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Explicit triples of isomorphic Riemann surfaces
Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....
3
votes
0
answers
129
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Action of the mapping class group on curves and triangulations
Consider an orientable surface $S$ of arbitrary genus, possibly with boundaries, and with marked points and/or punctures. I will assume that every boundary has at least one marked point so that the ...
4
votes
1
answer
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Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees
I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \...
4
votes
2
answers
195
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Measured geodesic laminations have either discrete or Cantor set local cross-sections
I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...
1
vote
1
answer
122
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Inner product on global sections of positive line bundle
Let $\Sigma = S^2$ be thought of as a Riemann surface, and let $L$ be a Hermitian line bundle on $\Sigma$ with curvature $2$-form $-2 \pi i \Omega \in \Omega^2(\Sigma, \mathbb{R})$. Then $L$ is a ...
2
votes
0
answers
94
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Vortex equation on Riemann surface and a similar equation
Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...
6
votes
1
answer
378
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When is a compact orbifold Riemann surface a global quotient of a Riemann surface
While reading the paper Seifert Fibred Homology 3-Spheres and the Yang-Mills Equations on Riemann Surfaces with Marked points by M. Furuta and B. Steer, I stumbled upon the following statement:
Any ...
5
votes
2
answers
412
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Fenchel-Nielsen length-length coordinates on Teichmueller space?
Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...
2
votes
0
answers
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Triangulations with discrete metrics and conformal equivalence
A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...
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0
answers
130
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definition of generic function
what is definition of generic function in following paper ? i need a reference for definition generic function .
"A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed ...
2
votes
1
answer
208
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Is action of MCG on the curve complex computable for closed surfaces? [Yes: Birman Exact Sequence]
$\DeclareMathOperator\MCG{MCG}$Let $\Gamma=\MCG(S_{g, 0,0 })$ be mapping class group of closed hyperbolic surfaces. Let $V=C^0(S)$ be the set of vertices of the simplicial curve complex. We are ...
7
votes
2
answers
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Do surface groups embed into PSL_2 over a real quadratic integer ring?
$\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be ...
1
vote
1
answer
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operations on matrices preserving the property of being the Riemann matrix of a surface
I have heard about the Schottky problem and the related Novikov's conjecture about the characterization of matrices in the Siegel upper half-space which are indeed the Riemann matrix of a compact ...
16
votes
1
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843
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Proving algebraicity of compact Riemann surfaces without Chow's theorem
I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
2
votes
1
answer
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Can we always find coordinates on a surface such that $K=K(u-v)$?
Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function ...
6
votes
3
answers
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Graphs from the point of view of Riemann surfaces
I was listening to the lecture "Graphs from the point of view of
Riemann surfaces" by Prof. Alexander Mednykh. I am looking for references for the basics of this topic. Any kind of ...
3
votes
1
answer
118
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Nonrepresentability by radicals and entire (or meromorphic) functions of algebraic functions
It is known that an algebraic function with non-solvable monodromy group can not be represented by radicals. Where can we find a detailed proof about the nonrepresentability by radicals and entire (or ...
4
votes
1
answer
477
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The real part of the period of an elliptic curve
Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \...
1
vote
0
answers
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Simplest Liouville Manifold not of Finite Type, or - Liouville Cobordism Structure on Pair of Pants?
I've been trying to produce the simplest possible example of a Liouville manifold which wouldn't be of finite type (a Liouville manifold is said to be of finite type if its skeleton is compact), and ...
2
votes
1
answer
197
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A Riemann surface is automatically paracompact
[A question I remember from many years ago.]
Definition
A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...
3
votes
1
answer
215
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Schwarzian derivative, accessory parameters, projective connections
I am looking at the following Riemann surface (let's call it $M$),
\begin{equation}
y^n=\frac{(x-x_1)(x-x_3)}{(x-x_2)(x-x_4)}
\end{equation}
which is a Riemann surface of genus $n-1$. It can be ...
4
votes
1
answer
284
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Is there a decision procedure for analytic continuation?
Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of ...
5
votes
0
answers
129
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Selberg zeta function analytic expressions
Consider the following algebraic equation,
$$
y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}
$$
which is a Riemann surface of genus $n-1$ (after compactifying). The classical retrosection theorem due to ...
4
votes
1
answer
366
views
Reverse residue theorem without using Serre's duality
In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text):
Let $\{a_1, \dots,a_n\}$ be a set of points in ...
1
vote
1
answer
62
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Existence of continuous family of uniformising parameters
I asked this question on MSE a while ago but didn't receive any useful answers.
Suppose I have a $1$-parameter family continuous maps $f_t: \mathbb{S}^2\rightarrow \mathbb{C}P^1$ from a topological $2$...
2
votes
1
answer
214
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Two definitions of Teichmüller space: relative isotopy or not?
The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism.
The definition on ...
4
votes
0
answers
145
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Products of eigenfunctions on compact Riemann surfaces
Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
1
vote
0
answers
46
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Uniformization of triangulation on a sphere up to Moebius transformations
This is not the most precise question but rather a hope that someone has seen something like this.
I am given a triangulation of the 2-sphere $S^2$ which I only know up to Moebius transformations. I ...
4
votes
1
answer
878
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Laplace-Beltrami of the mean curvature
For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...
10
votes
1
answer
923
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Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles
I am trying to visualize the genus-two Riemann surface given by the curve
$$
y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}.
$$
We can regard this surface as a three-fold cover of the sphere with four ...
5
votes
2
answers
506
views
Explicit example de Rham moduli space of connections
Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:
-$M_{Dol}$ the moduli space of stable ...
1
vote
1
answer
298
views
Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials
According to Riemann surfaces, dynamics and
geometry
by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by
$$
\|\phi\|_p = \left(\...