# 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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### Lengths of generators of surface group

Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...

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### Branched covering maps between Riemann surfaces

What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!

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### When do two measured foliations on a surface define a Riemann surface structure?

Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...

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### Simple smooth functions on Bolza surface

Consider the Bolza surface, a compact Riemann surface of genus 2.
It is an octagon in the Poincaire disk model with opposite sides identified.
I would like to write down some analytic expressions for ...

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### Reference request: uniformization theorem proof by Borel

This answer refers to a proof of the uniformization theorem via the PDE describing metrics of constant curvature (Liouville?) by Borel. I haven’t been able to find this reference, is anyone aware ...

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### Are Bergman metrics on compact Riemann surfaces continuous on Teichmüller space?

Let $R$ be a compact Riemann surface of genus $\geq 1$, and let $\omega_1,\ldots,\omega_g$ be holomorphic one forms that form the dual basis of canonical homology basis. Let $(\pi)_{ij}$ be the ...

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### Making a map in sheaf cohomology involving a theta characteristic explicit

Motivation:
For a given rank 2 vector bundle we want to know how many theta-characteristic valued twisted endomorphisms it has.
Setting:
Let $C$ be a smooth algebraic curve over a field of ...

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### Symplectic form on the space of geodesic currents on a surface?

There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...

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### Holomorphic fibre bundles over noncompact Riemann surfaces

Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...

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### On nontrapping manifolds

Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary.
Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...

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### Teichmüller theory for open surfaces?

I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...

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### Reference for Teichmuller spaces of punctured surfaces

What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$?
I am looking for a reference where there is the correct statement and or proof of say the Bers embedding,...

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### Serre duality for non-compact Riemann surfaces

Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology
$$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$
Can we say ...

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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 $x,y$.
...

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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 ...

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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 \...

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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 ...

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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 ...

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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 ...

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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 $g$ ...

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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}(\...

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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 ...

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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})}}...

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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 $\...

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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 $\...

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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 $c \in \mathbb C$...

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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 ...

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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 ...

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### 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 ...

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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?

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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_{...

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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 ...

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### 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 : ...

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### 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 ...

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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 ...

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### 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, $L_X g = (\mathrm{div} X) g$ ...

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### 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 $X$ ...

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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 ...

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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. ...

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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 ...

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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) ...

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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+...