All Questions
Tagged with cv.complex-variables riemann-surfaces
10 questions
31
votes
11
answers
13k
views
Uniformization theorem for Riemann surfaces
How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
29
votes
7
answers
7k
views
Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
user100272
20
votes
2
answers
1k
views
Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $...
- 6,548
9
votes
3
answers
927
views
Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes.
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always ...
- 5,590
12
votes
2
answers
849
views
Visualizing holomorphic differentials on a compact Riemann surface?
It is a classical result that the vector space of holomorphic differentials on a compact Riemann surface of genus $g$ has dimension $g$. I am wondering if there is a way of visualizing this wonderful ...
- 82.6k
12
votes
2
answers
2k
views
Universal covering of a 2-sphere without $n$ points
Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...
- 21.8k
11
votes
3
answers
748
views
Explicit triples of isomorphic Riemann surfaces
Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....
9
votes
3
answers
1k
views
An analytic proof of the De Franchis theorem
The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...
- 1,159
8
votes
2
answers
392
views
Image of boundary circle under map from punctured elliptic curve to ℂ
Let $E=\mathbb C/\Lambda$ be an elliptic curve,
and let $D\subset E$ be a very small disc.
($D$ is round for the usual flat metric on $E$)
By the main result of [1], there exists a holomorphic ...
- 43.2k
7
votes
1
answer
794
views
Non-algebraic curve visualisation
Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert ...
- 1,343