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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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points of $X$ with non trivial stabilizers are discrete

so far I understand about the statement: let $p_i,i=1\dots,n$ has non trivial stabilizers i.e $S_{p_1}=\{g:g.p=p, g\in G\}\neq\{e\}$, is non trivial subgroup of $G$ for $p_1$ and so forth upto $p_n$ we will get $S_{p_n}$,so we need to show…
Myshkin
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Every rational function which is holomorphic on Riemann Sphere($\mathbb{C}_{\infty}$)

could any one give me a hint how to show Every rational function which is holomorphic on every point of Riemann Sphere( $\mathbb{C}_{\infty}$) must be constant?(with out applying Maximum Modulas Theorem). Thank you.
Myshkin
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map from $D$ to $\pi(D)$ is injective?

please follow this map from $\mathbb{C}$ to $\mathbb{C}/L$ is open map? ,let w be a non zero element of the lattice L so that |w|>2ϵ, fix such ϵ>0 and any $z_0$∈C and take an open disk of radius ϵ with centre at $z_0$, could you please tell me why…
Myshkin
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why this map is injective?

let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in\mathbb{C}^{*}$ is a point in $\mathbb{CP}^1$, the map is $\phi:U_0\rightarrow\mathbb{C}$ defined by $$\phi([z:w])=w/z$$…
Myshkin
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Is the Coordinate Chart Always a Biholomorphism?

Let $X$ be a Riemann surface with complex structure $\{(U_i,\phi_i)\}$. Is it the case that $\phi_i:U_i\rightarrow V_i$ is a biholomorphic map in the sense of Riemann surfaces?
Edward Hughes
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Under what conditions do there exist non-constant meromorphic functions between general Riemann surfaces?

The uniformization theorem answers this question for particular Riemann surfaces, but do we have a general theorem for this? Do we also get meromorphic functions that can separate points?
user98246
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Why $\mathbb C/ \Lambda$ is a Riemann surface?

Consider a complex lattice $\Lambda \subset \mathbb C$ generated by some real basis $\{w_1,w_2 \}$. I want to know why the quotient group $\mathbb C/ \Lambda$ has a unique structure as Riemann surface.
SamC
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relation between degree and residues

Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set $$V=H^0(C,\omega_C(D))$$ the set of section of $\omega$ with poles…
dario
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1-forms on Riemann surface with boundary

Suppose $R$ is orientable compact Riemann surface with boundary $\partial R$ a collection $C_1, \dots C_k$ of oriented circles, so that for $i=1,\dots,j$, $C_i$ has boundary orientation, and for $i=j+1,\dots k$, $C_i$ has nonboundary orientation.…
Yann
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