# Tagged Questions

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**1**answer

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### A question for hyperbolic metric in the proof for Bohr's lemma

Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question:
Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on ...

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109 views

### Use of Jensen's inequality on a Riemann surface

Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square ...

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82 views

### Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?
Thanks,

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68 views

### How to find number of points at infinity of a Riemann surface

Let $X \subset \mathbb C^2$ be a Riemann surface with boundary $\partial X \subset \mathbb C^2$ and without compact components. Let $\bar X = X \cup \{p_1,\ldots,p_N\} \subseteq \mathbb CP^2$ be its ...

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114 views

### Residues and Mittag-Leffler sequence

Let $X$ be a compact Riemann surface, $\omega$ a meromorphic differential on $X$ and $f$ a meromorphic function on $X$ with poles only over the points $P_1,\dots,P_d$. The product $\;f\cdot\omega\;$ ...

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**1**answer

147 views

### Approximation Runge's Theorem

Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component ...

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123 views

### Trivial Line Bundle-Riemann surfaces

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...

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204 views

### Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be:
-Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function.
-PoincarĂ© hyperbolic if it is covered by the unid disk.
Are this ...

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**3**answers

393 views

### Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$.
Is $X - \overline{D(x,r_x)}$ hyperbolic?

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533 views

### Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that ...

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412 views

### A “holomorphic” Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.
I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...

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338 views

### Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we ...

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295 views

### Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...

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209 views

### Is the absolute value of the j-invariant bounded from below on an annulus

Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip ...

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793 views

### Motivation behind defining the Ramification Divisor

I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function.
As I see the definition,
If $f$ is a meromrophic function between two ...

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**4**answers

557 views

### $Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]

I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...

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596 views

### Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...

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**1**answer

518 views

### Non-algebraic curve visualisation

Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert ...

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**1**answer

270 views

### Necessary condition for a branch point

If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend ...

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1k views

### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...

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1k views

### Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following:
1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...

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1k views

### Branched coverings of Riemann surfaces with specified branch points.

Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...