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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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Unramified holomorphic map is isomorphism

In my course about Riemann surfaces, the professor briefly mentioned the following as a fact that we shall just accept: If $X$ is a connected, compact Riemann surface and $f:X\to\mathbb C_\infty$ is a holomorphic unramified map, then $f$ is…
Zuy
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Equivalence of Weierstrass and Ramification points of a Riemann Surface

It is known that on a hyperelliptic surface the set of Weierstrass points and the set of ramification points of the extension of the projection map $(x,y)\mapsto x$ to $\mathbb{P}^1$ coincide. However, I am not sure if this is the case for a general…
AnonymousCoward
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Unramified map to a simply connected surface

I want to know why if f is an unramified covering to a simply connected surface then this function is an isomorphism?
Mathgreek
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Curve lifting property

I'm reading forster's riemann surfaces book and I read this theorem: if we have a covering map between two topological spaces X,Y then it has curve lifting property, now I want to make a counter example for the inverse,but I couldn't find. Why…
Mathgreek
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Surface Area of Curve about y-axis

I'm trying to rotate the curve $$ \frac{1}{4} x^{2}-\frac{1}{2} \ln x $$ with $$ 1 \leq x \leq 2 $$ about the y-axis. I know the formula for this is: $$ \int_{a}^{b}\left(2 \pi f(y) \sqrt{1+\left(f^{\prime}(y)\right)^{2}}\right) d y $$ The only…
ganondork
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an example for sheaves

I want to know whether we can define a sheaf for holomorphic functions or not,I check the axioms of when a presheaf could be a sheaf,but I'm not sure it is correct.for axiom 1 I use identity theorem and for axiom 2 we should say there exist a…
Mathgreek
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Poles of a meromorphic function on riemann sphere is finite

By identity theorem,I want to prove that every meromorphic function $f$ on riemann sphere $\mathbb{P}^1$ has finitely many poles, can this solution be true? : Suppose $f$ has infinite poles then the restriction…
Mathgreek
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Multiplicity of a complex polynomial in $\mathbb{P}^1$

This is based on Example 2.3 on Forster's "Lectures on Riemann Surfaces": let $f(z)=z^k+c_1z^{k-1}+...+c_k$ be a complex polynomial of degree $k$. Then $f$ can be considered as a holomorphic mapping $f:\mathbb{P}^1\rightarrow\mathbb{P}^1$ where…
Marra
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Fixed points of an automorphism and a meromorphic map.

I want to solve this problem Let $X$ a compact Riemann surface and $f:X\rightarrow \mathbb{P}^1$ a non constant meromorphic function in $X$. Let $h\in\text{Aut }(X) $ and automorphism of finite order $\text{ord } (h)$. If the number of fixed points…
EQJ
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Torus and period integrals

I'm following a course in Riemann surfaces, and I'd like to solve the exercise below. Let $L$ a lattice in $\mathbb{C}$, and let $T:= \mathbb{C}/L$ the corresponding torus. i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the…
Pippo
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Sketch of proof of Uniformization Theorem

Can somebody give me the sketch of the proof of the uniformization theorem, given that every compact Riemann surface admits a non-constant meromorphic function. (I am not so sure these two are indeed related or not.) I just want to know how the…
Mike Park
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Are Cartesian coordinates unable to explore multi-valued functions beyond the principle value?

By expressing the domain of a function in polar coordinates it is possible to take the function beyond it’s principle value (ie beyond 2$\pi$) to reveal the full beauty & complexity of it’s Riemann surface. Am I right in saying that Cartesian…
Peter Smallwood
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Diffeomorphism between $X$ of genus $g>1$ and the complex projective line $\mathbb{CP}^1$

Is it automatic that a compact Riemann surface $X$ of genus $g>1$ is not homeomorphic to a compact subset of the complex projective line $\mathbb{CP}^1?$ Note: I apologize for the confusion-I am just a high school student with an interest in math.…
Sergio Charles
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How are the following two definitions of holomorphic mappings on Riemann surfaces equivalent?

A holomorphic function $\phi:X\to Y$, where $X$ and $Y$ are Riemann surfaces, is described in the following way: For $a\in X$, let $a\in U_1$, where $U_1$ is an open set. Let $C_1:U_1\to V_1$ be a chart. Similarly, let $f(a)\in U_2$, and let…
user67803
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A question about the gluing of Riemann surfaces

Often, the definition of Riemann surfaces is motivated by the example of the multi-valued function $f(z)=\sqrt{z}$. Every point $z\in \Bbb{C}$ has two images. Hence, this function has two "branches"; $re^{i\theta}\to \sqrt{r}e^{i\theta/2}$ and…
freebird
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