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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Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$. ...
stupid_question_bot's user avatar
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Coupling small and large injectivity radii

I'd like to know whether a manifold of constant curvature, which has large injectivity radius at many points, can have points of arbitrary small injectivity radius. More precisely, for a point $x$ in ...
Nandor's user avatar
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Bound on the sum of intersection number of any projectivized measured foliation with two transverse measured foliations

Let $R$ be a finite Riemann surface (having negative Euler Characteristic) without boundary (may have punctures) and $q$ be a unit area quadratic differential on $R$. We define $\mathcal{MF}_{1}=\{F \...
P.S's user avatar
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Čech cohomology refinement mapping

Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...
Alexander Mrinski's user avatar
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Relations between two definitions of harmonic measure

I came into two definitions of harmonic measure on a Riemann surface. The first is defined on p.180 of Riemann surfaces, 2nd by Kra and Farkas, which read as follows. Theorem. Let $M$ be a hyperbolic ...
gaoqiang's user avatar
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A compact Riemann surface with a finite set of points removed is parabolic

A Riemann surface $\mathcal{R}$ is called parabolic if it is not compact and doesn't carry a negative non-constant subharmonic function, and is called hyperbolic if it carries a negative non-constant ...
gaoqiang's user avatar
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Hyperelliptic integrals

I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
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Conformally embedding a finite Riemann surface of genus g

Let $R$ be a compact Riemann surface of genus $g$ and let $S \subset R$ be a Riemann subsurface. Theorem B in Maskit's paper says that we can embed $S$ into a compact Riemann surface $P$ of genus $g$ ...
Jaikrishnan's user avatar
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Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism

This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand: $$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
Kenny S's user avatar
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Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title: The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
Chaitanya Tappu's user avatar
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Elliptic integral as quantity associated with Riemann surface?

There are many elliptic integrals, so to show my point let me just pick one of them (complete elliptic integral of the first kind [1]): $$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
Student's user avatar
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Gonality of specific Riemann surfaces $y^k=\tfrac{z^k-1}{z^k+1}$

The gonality of a compact Riemann surface $\Sigma$ is defined to be the lowest degree $d$ of a non-constant holomorphic map $f\colon \Sigma\to\mathbb CP^1.$ This means the gonality is 1 only for $\...
Sebastian's user avatar
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Area of a smooth complex projective curve

Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\...
Daniel Asimov's user avatar
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Which holomorphic curves can be leaves of a non-singular holomorphic foliation of $\mathbb C^2$?

It is easy to see that for any entire function $f : \mathbb C \to \mathbb C$, its graph $G(f) = \{(z,f(z)) \in \mathbb C^2 \mid z \in \mathbb C\}$ can be translated by $(0,c)$ for any $c \in \mathbb C$...
Daniel Asimov's user avatar
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When does a group act effectively and holomorphically on some Riemann surface?

Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary ...
Matthew Niemiro's user avatar
5 votes
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Gaussian curvature of a holomorphic curve in complex 2-space

Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$. Each point of $M$ has ...
Daniel Asimov's user avatar
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1 answer
274 views

Realizing a finite subgroup of $\mathrm{Homeo}^+(S_g)$ as a subgroup of $\mathrm{Isom}^+(S_g)$

Let $G\leq \operatorname{Homeo}^+(S_g)$ be finite, where $S_g$ is a closed, connected, orientable surface of genus at least $2$. Then I have the following questions: (1) Can $G$ always be realized as ...
Rajesh Dey's user avatar
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Uniformization of $\mathbb{CP}^2-\bigcup C_i$, where $C_i$ are Riemann surfaces intersecting generically

Consider $X=\mathbb{CP}^2-\bigcup C_i$ where $C_i$ are Riemann surfaces intersecting generically. How to compute the fundamental group of this space and what is the universal cover?
0x11111's user avatar
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Canonical basis of cycles of Riemann surfaces

Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve $$ f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0, $$ where $a_1(x), \dots, a_n(x)$ are ...
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Size of conformal factor under uniformisation

Consider closed orientable surfaces whose metrics are hyperbolic (i.e., $K=-1$) except in a region which is a hemisphere of a unit sphere attached to the hyperbolic region along a closed geodesic (of ...
Mikhail Katz's user avatar
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Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?

Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$? Motivation: I had intention to consider this question ...
Ali Taghavi's user avatar
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98 views

Geodesics in free homotopy classes and the fundamental group

Let $\mathcal{H}$ be the upper half-plane and $\Gamma$ be a cocompact, torsion-free Fuchsian group. The quotient space $X=\Gamma\backslash \mathcal{H}$ is a smooth closed Riemann surface and there is ...
Claudius's user avatar
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2 answers
325 views

Holomorphic Gauss normal map

Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$. Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic ...
Ali Taghavi's user avatar
2 votes
0 answers
76 views

Two different Bers embeddings

In An Introduction to Teichmüller spaces by Imayoshi and Taniguchi, they present in section 6.1.3 the Bers embedding as a map from Teichmüller space of a Riemann surface $X$ to the space of quadratic ...
Jacques's user avatar
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How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?

Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation: $$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
Rajesh Dey's user avatar
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A couple of questions about the moduli space of annuli with some marked points on the boundary components

I'm trying to work out an answer for my previous question and I'm stuck with the following issue: In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
Riccardo's user avatar
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6 votes
1 answer
391 views

Existence of a holomorphic map between Riemann surfaces

Nevanlinna in his book Analytic functions seems to state the following (at the very end of Ch. X): For every compact Riemann surface $X$ of genus $g\geq 2$ there is a non-constant holomorphic map $f : ...
Alexandre Eremenko's user avatar
4 votes
1 answer
188 views

Conformal map between flat and hyperbolic torus with a boundary

I am confused because I can define two very different complex structures on the torus with a puncture/boundary. For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
Holomaniac's user avatar
1 vote
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Unexpected holomorphic tubular neighborhood

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
Mohan Swaminathan's user avatar
5 votes
2 answers
331 views

Conformal Killing vector fields on compact surface of genus \ge 1

Let $(M, g)$ be a compact 2-dimensional Riemannian manifold with genus $\ge 1$. Can $M$ has a conformal Killing vector field $X$ other than Killing vector fields? That is, $L_X g = (\mathrm{div} X) g$ ...
user486255's user avatar
2 votes
1 answer
66 views

Detecting non-affine automorphisms of a translation surface

Let $(X, \omega)$ be a translation surface, i.e., a Riemann surface with a homomorphism $1$-form. A central object is the group of affine automorphisms $\text{Aff}^+(X, \omega)$: homeomorphisms of $X$ ...
Sam Freedman's user avatar
1 vote
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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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1 vote
1 answer
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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
1 vote
0 answers
109 views

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_{...
zapkm's user avatar
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3 votes
2 answers
445 views

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
2 votes
0 answers
226 views

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
1 vote
1 answer
165 views

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
0 answers
95 views

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
455 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
296 views

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
5 votes
0 answers
251 views

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
  • 545
2 votes
1 answer
146 views

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
0 answers
63 views

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; ...
TSBH's user avatar
  • 101
5 votes
0 answers
126 views

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
  • 1,654
2 votes
1 answer
106 views

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
209 views

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
  • 379
4 votes
2 answers
476 views

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
  • 4,045
2 votes
0 answers
278 views

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
  • 222
2 votes
0 answers
265 views

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
376 views

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 ...
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