Questions tagged [riemann-surfaces]
For questions about Riemann surfaces, that is complex manifolds of (complex) dimension 1, and related topics.
2,259 questions
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Why is the point at infinity so natural in complex analysis but not in real analysis?
I am trying to understand the conceptual role of the “point at infinity” in complex analysis and why it seems much more natural and useful there than in real analysis. In topology, one can always take ...
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Can the holomorphic functional calculus lead to some interesting algebraic geometry?
I was just introduced to the holomorphic functional calculus on Banach algebras. It was a light introduction, and the meromorphic case wasn't discussed, but I promptly found out that there also exists ...
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Riemann surfaces diffeomorphic to $\mathbb{C} \setminus \{ 0,1\}$
Consider the smooth manifold $\mathbb{C} \setminus \{ 0,1\}$ or "twice punctured plane" or "thrice punctured sphere". (Note that this is not the space called "pair of pants&...
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Second fundamental form and geodesic curvatures of frame lines
Consider a smooth surface $S$ in $\mathbb{R}^3$ (assuming simple connectivity if necessary) that is parametrized by $\mathbf{r}(u,v)$. With the second fundamental form written as $\text{II}=L\:\mathrm{...
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In what sense is Riemann surface of $\sqrt{z}$ not just $\mathbb{P}^1$?
This is an intentional duplicate of the question Why is the Riemann Surface of $\sqrt{z}$ not just $\mathbb{P}^1$, which was never answered. The Riemann surface of $f(z)=\sqrt{z}$, defined as:
\begin{...
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"Portals" in $\mathbb{C}$: branch cuts that translate function arguments
First some background and motivation.
I recently came across a YouTube video where the creator tried to calculate the effects of portals on the gravitational field (although electrostatics should work ...
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A problem of orientations when making use of Stokes' Theorem to prove a theorem on Riemann surface
This is a proof of one theorem from Terrence Napier and Mohan Ramachandran's An Introduction to Riemann Surfaces. I found some references which provide a fact that there is only one smooth structure ...
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degree of coordinate function on a hyperelliptic riemann surface
I am wondering why my computation for the degree of a coordinate function on a hyperelliptic Riemann surface is incorrect.
My definition of a hyperelliptic Riemann surface is as follows: Let $h(z)$ be ...
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“Algebraic” Abel-Jacobi theorem
Let $X$ be a compact connected Riemann surface and $x_0$ be a point of $X$. Abel-Jacobi theorem asserts that there is an isomorphism $Div^0(X)/PDiv(X) \to H^0(X, \Omega)^\vee/H_0(X,\mathbb{Z})$ ...
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How to prove that the moduli space $\mathcal{M}_g$ is Hausdorff using Kuranishi families?
Let $ C $ be a compact connected Riemann surface of genus $ g > 1 $. I used the definition of a Kuranishi family of $ C $ as in [Teichmüller space via Kuranishi families] 1.
Using these Kuranishi ...
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Question if a lemma from Miranda is classical u-sub
In Miranda’s Algebraic Curves & Riemann surfaces, in chapter IV. Integration on manifolds, Lemma 3.9(f) reads:
If $F:X\to Y$ is a holomorphic map between Riemann surfaces, then the operation (push ...
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Problem out of Miranda on Zero mean theorem
Let $X$ be a compact Riemann surface and suppose the zero mean theorem holds:
Zero mean Theorem (p.318 Miranda):If $X$ is an algebraic curve and $\eta$ is a $C^\infty(X)$ $2-$form on it,
then there ...
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Miranda X.$2$.F.
Let $X$ be a compact Riemann surface. Let Bar: $\Omega^1(X)\to H^{(0,1)}_{\bar{\partial}}(X)$
by sending $\omega$ to the equivalence class of $\bar{\omega}$ is $\Bbb{C}-$linear, and $1-1$.
Attempt:
So ...
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IV.I.$1$ out of Miranda Algebraic curves & Riemann Surfaces.
Let $L$ be a lattice in $\Bbb{C}$, and let $\pi$ be the natural protection. Show that $dz,d\bar{z}$ are well-defined holomorphic $1-$forms on $X=\Bbb{C}/L$.
Attempt:
I used charts because 2 are enough ...
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On scattering theory of Riemann Surfaces
In this 2025 paper on Scattering theory Scattering theory on Riemann Surfaces, I have some technical questions when tying it together with what I have learned in my courses. Let $R$ be a compact ...
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