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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 ...
Josh Lam's user avatar
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2 votes
1 answer
121 views

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!
cata's user avatar
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1 answer
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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,\...
W.Smith's user avatar
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1 vote
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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 ...
nervxxx's user avatar
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1 answer
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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 ...
Alex Bogatskiy's user avatar
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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 ...
François Fillastre's user avatar
2 votes
1 answer
203 views

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 ...
clemens_nollau's user avatar
9 votes
0 answers
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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 ...
Ian Agol's user avatar
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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 ...
Don's user avatar
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1 answer
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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 ...
Ali's user avatar
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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 "...
M.G.'s user avatar
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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,...
Chitrabhanu's user avatar
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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 ...
Aidan's user avatar
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0 answers
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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
2 votes
1 answer
90 views

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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1 answer
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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 \...
W.Smith's user avatar
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1 answer
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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
2 votes
1 answer
145 views

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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3 votes
1 answer
250 views

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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1 answer
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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 ...
mxjia's user avatar
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0 answers
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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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2 votes
1 answer
217 views

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
4 votes
0 answers
100 views

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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4 votes
1 answer
184 views

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
  • 6,755
12 votes
3 answers
966 views

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
3 votes
0 answers
118 views

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
11 votes
2 answers
1k views

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
2 answers
383 views

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
5 votes
1 answer
279 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
2 votes
1 answer
375 views

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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1 vote
0 answers
92 views

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 ...
mxjia's user avatar
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1 vote
0 answers
49 views

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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4 votes
2 answers
385 views

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
0 votes
0 answers
128 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
  • 218
4 votes
2 answers
346 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
89 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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2 votes
1 answer
258 views

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
4 votes
0 answers
87 views

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
427 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
194 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
0 answers
214 views

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
436 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, $L_X g = (\mathrm{div} X) g$ ...
Sean's user avatar
  • 149
2 votes
1 answer
73 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
0 answers
84 views

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
  • 41
2 votes
1 answer
340 views

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
121 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
  • 541
3 votes
2 answers
476 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
242 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
167 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

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