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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Metric balls in Teichmüller space are topological balls

Let $X$ be a topological surface of finite type and $\mathcal{T}_X$ be the corresponding Teichmüller space. Let $B$ be a ball with respect to the Teichmüller metric on $\mathcal{T}_X$ (i.e., the ...
A B's user avatar
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Pythagorean theorem in Riemann metrics of non constant curvature

I already asked the same question here, but received no answer. I was reading this interesting article by Givental Givental, Alexander. "The Pythagorean theorem: What is it about?" Amer. ...
user967210's user avatar
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Existence of meromorphic one-form with a fixed order pole

Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{...
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Finding a hyperbolic metric with geodesic boundary on a given Riemann surface

Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics ...
Yuxiao Xie's user avatar
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A Question about an article by Birman, Series

Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
Amirhossein's user avatar
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1 answer
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Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$

Bring's curve or Bring's surface with genus 4 and $5!=120$ automorphisms can be given by the homogeneous equations, $$x_1+x_2+x_3+x_4+x_5 = x_1^2+x_2^2+x_3^2+x_4^2+x_5^2 = \\x_1^3+x_2^3+x_3^3+x_4^3+...
Tito Piezas III's user avatar
3 votes
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A generalisation of Cauchy-Stieltjes transform

For a nice function $\nu$ (say smooth and compactly supported), its Cauchy-Stieltjes transform is defined as $$\int_\mathbb R \frac{\nu(s)}{z-s}\mathrm{d}s$$ which is holomorphic in $\mathbb C\...
Jiyuan Zhang's user avatar
2 votes
3 answers
369 views

Groups of conformal isomorphisms of simply connected surfaces

By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces: open disk $D$, complex plane $\mathbb{C}$, or $2$-...
Sergiy Maksymenko's user avatar
5 votes
3 answers
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Classification of surface bundles over surfaces

Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces? Now, if the fibre F and the base B are both ...
Daniel Asimov's user avatar
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What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?

$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...
C.D.'s user avatar
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Curvature of curves through a point of a surface smoothly embedded in Euclidean space

The curve C(𝜃) drawn on a smoothly embedded surface 𝜮 in 3-space — where C(𝜃) is defined as the intersection of 𝜮 with a 2-plane perpendicular to 𝜮 at P — leaving the point P at angle 𝜃 will ...
Daniel Asimov's user avatar
1 vote
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Computing some closed trajectories of meromorphic quadratic differentials

I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; ...
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Algebraic dependence of the elliptic functions

Let $\{f_i\}_{i=1}^{n}$ be $n$ elliptic functions in $\mathbb{C}$. We say that $f_1, \dots, f_n$ are algebraic dependent over $\mathbb{C}$ if there exists a polynomial of $n$ variables with constant ...
yaoxiao's user avatar
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References for group of invariance of the Painlevé property

I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
Redouane Khaled's user avatar
5 votes
1 answer
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Which punctured Riemann surface are the complex structures of complete minimal surfaces in $\mathbb{R}^3$?

Question: Let $\Sigma$ be a punctured Riemann surface(i.e. a closed Riemann surface with several points removed). Is there always a complete conformal minimal immersion $X: \Sigma \to \mathbb{R}^3$? ...
gaoqiang's user avatar
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4 votes
2 answers
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On diffeomorphisms that preserve the metric

Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that $$ F: \Omega \to \Omega,$$ is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is ...
Ali's user avatar
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2 votes
0 answers
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Uniformization of Riemann surfaces with cone singularities

Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such ...
Josh Lam's user avatar
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Monodromy action

Let us consider the following tower of (finite) ramified Galois covers $$S \xrightarrow{p} \mathbb{P}_1 \xrightarrow{q} \mathbb{P}_1,$$ where $S$ is a Riemann surface. Denote by $R \subset \mathbb{P}...
user494203's user avatar
8 votes
2 answers
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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 ...
gaga's user avatar
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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 ...
user494203's user avatar
7 votes
1 answer
245 views

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 ...
André Henriques's user avatar
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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 ...
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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{...
dennis's user avatar
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1 answer
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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 ...
AMHG's user avatar
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1 answer
459 views

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 - ...
dennis's user avatar
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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 ...
Sidharth Ghoshal's user avatar
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1 answer
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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} \...
P.S's user avatar
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1 answer
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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 ...
Lukasz Kosinski's user avatar
3 votes
1 answer
355 views

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 ...
gaoqiang's user avatar
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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,...
Sam Freedman's user avatar
8 votes
1 answer
311 views

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 ...
math101's user avatar
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4 votes
0 answers
212 views

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 ...
Victor Felipe's user avatar
3 votes
1 answer
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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 ...
Daniel Asimov's user avatar
4 votes
0 answers
136 views

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 ...
tota's user avatar
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3 votes
1 answer
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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 ...
giulio bullsaver's user avatar
2 votes
1 answer
173 views

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 ...
Pène Papin's user avatar
11 votes
3 answers
700 views

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
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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 ...
giulio bullsaver's user avatar
4 votes
1 answer
118 views

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 \} \...
nandi's user avatar
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1 vote
1 answer
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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 ...
Harry Reed's user avatar
1 vote
1 answer
109 views

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 ...
skr's user avatar
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1 vote
0 answers
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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 ...
Partha's user avatar
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6 votes
1 answer
307 views

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 ...
Akerbeltz's user avatar
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5 votes
2 answers
301 views

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 ...
JHM's user avatar
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2 votes
0 answers
51 views

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 ...
Andi Bauer's user avatar
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1 vote
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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 ...
Usa's user avatar
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2 votes
1 answer
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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 ...
JHM's user avatar
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7 votes
2 answers
341 views

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 ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
52 views

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 ...
azbuk8's user avatar
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15 votes
1 answer
680 views

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 ...
Jas Singh's user avatar
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