# Tagged Questions

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### The number of holomorphic coverings (with given degree) of the punctured sphere is finite.

I'm looking for a proof of the following theorem: Fix a finite set $B=\{y_1,\ldots,y_k\}\subseteq \mathbb P^1(\mathbb C)$, then there is only a finite number of isomoprhism classes of ...
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### References for a standard result about coverings of Riemann surfaces

I my thesis I have to cite the following standard result: Let $Y$ be a compact Riemann surface and let $B\subseteq Y$ be a finite subset. Given a natural number $d$, there are only finitely many ...
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### Are two equivalent coverings of a Riemann surface also biholomorphic?

Consider a compact Riemann surface $X$. If $p_1:Y\longrightarrow X$ and $p_2:Z\longrightarrow X$ are two topological coverings of $X$, then it is univocally defined a structure of Riemann surface on ...
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### Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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### Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$

I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...
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### Constructing Riemann surfaces using the covering spaces

In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow: A polynomail-like map ...
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### Monodromy Theorem and Homotopy Lifting Theorem

I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct? Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
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### Calculating monodromy

I'm right now learning about Monodromy from self-studying Rick Miranda's fantastic book "Algebraic Curves and Riemann surfaces". Today, I read about monodromy, and the monodromy representation of a ...