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Questions tagged [riemann-surfaces]

For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.

Riemann surfaces are one-dimensional complex manifolds that are deformations of $\mathbb{C}.$

Riemann surfaces are orientable as a real manifold, and every simply connected Riemann surface is conformally equivalent to one of the following:

Elliptic (Positive curvature): The Riemann sphere (the complex plane with an extension to a point at infinity)

Parabolic (zero curvature): The complex plane

Hyperbolic (negative curvature): The open disk or upper half-plane

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Natural metric on unit disk

I am new to Riemann surface and I am struggling to understand the induced metric by the conformal structure. I am following the book Riemann Surface by Farkas and Kra and in the chapter IV.8 it talks about Riemannian metrics. It indroduces the…
Matteo Testa
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Rigidity of isometries of finite covers of Riemann surfaces

Let $\Theta$ and $\Sigma$ be compact Riemann surfaces with $\widetilde\Theta=\widetilde\Sigma=\mathbb{H}^2$ (so $\theta$ and $\Sigma$ are quotients of $\mathbb{H}^2$ by Fuchsian groups) such that $\rho:\Theta\rightarrow\Sigma$ is a finite cover…
Sam Hughes
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What does the Color and height of a Riemann surface represent

The title says it, but I think Color may represent angle and in part because magnitude, as height, is unbounded. Its frustrating that I haven't been able to find this from searching the web. Any answers are appreciated. EDIT: this is weirder than I…
Pineapple Fish
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Any two dimensional orientable manifold is complex

Let $M$ be a $2$-dimensional orientable manifold. Is $M$ a Riemann surface? If it is true, how can I show it? Thank you very much.
joseabp91
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Complex Structure on Riemann Surface

Is it always possible to give a complex structure on the one-point compactification of a Riemann surface such that compactification will be a Riemann surface? If it is possible, can we extend a proper holomorphic map between two Riemann surfaces to…
Sumanta
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X function of Riemann surface $y^3=(x-x_1)*(x-x_2)$

Im reading a book where they study the X function of the Riemann surface $y^3=(x-x_1)*(x-x_2)$ ($x, y \in \mathbb{S}^2$) and $dX$. They say that $dX$ has a zero of degree 2 in $x_1$ and $x_2$ and "clearly no other zero". However they don't give any…
Thomas
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Show that any harmonic map $f:\Sigma_1 \rightarrow \Sigma_2$ is a constant.

Let $\Sigma_1$ and $\Sigma_2$ be Riemann surfaces with nonnegative and negative curvature respectively. Show that any harmonic map $f:\Sigma_1 \rightarrow \Sigma_2$ is a constant. Thanks to the comment of Anthony Carapetis, it follows from Bochner…
gžd15
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Riemann sphere in hypercomplex number sets?

Do Riemann spheres work on bigger number sets than complex numbers? I think it would be interesting to think of quaternions as a 5D sphere or something, but I really don't know if the maths of Riemann spheres work in other dimensions. Thanks.
Okan Zagor A.
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Embedding of a Rieman surface of genus one in $\mathbb{P}^2$

In Jost's Compact Riemann Surfaces, he proves that every compact Riemann surface of genus one can be embedded into $\mathbb{P}^2$. Moreover, he proves the image of the embedding is the zero set of a polynomial of the form $y^2=x^3+ax+b$. I'm fine…
JeCl
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How to analytically continue this function?

I wish to show one solution of $w^{3}=z^{2}+1$ can be analytically continuation around $z=0,w=1$ to two different solutions. It seems the standard way is to consider the points $z=1,z=-1,z=\infty$ and connect them. Once the analytically continuation…
Bombyx mori
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genus of moduli space of a riemann surface

I am studying finite group action on Riemann surfaces from the book Algebraic Curves and Riemann Surfaces by Rick Miranda and there is are few statements about the genus that I am not able to understand: Let $X$ be a Riemann sphere and let $G$ is a…
user297008
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why the inverses of these maps are continuous?

$\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$. Let $(z,w)\in \mathbb{C}^2$ be non zero; its span is a point in $\mathbb{CP}^1$. Let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and…
Myshkin
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All compact genus 0 Riemann surfaces are isomorphic to a sphere

Where can I read the proof that all Riemann surfaces which are homeomorphic to a sphere are also isomorphic ?
Rupert
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$G$ be a group acting holomorphically on $X$

could any one tell me why such expression for $g(z)$, specially I dont get what and why is $a_n(g)$? notations confusing me and also I am not understanding properly, and why $g$ is automorphism? where does $a_i$ belong? what is the role of $G$ in…
Myshkin
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conformally equivalent riemann surfaces

Two riemann surfaces $S$ and $R$ are said to be conformally equivalent if there exist a holomorphic map $f:S\rightarrow R$ which is one-one and onto, and inverse is also holomorphic. I have to show No two of them are conformaly equivalent: $a$)…
Myshkin
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