All Questions
14 questions
2
votes
3
answers
478
views
Groups of conformal isomorphisms of simply connected surfaces
By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces:
open disk $D$, complex plane $\mathbb{C}$, or $2$-...
2
votes
1
answer
112
views
References for group of invariance of the Painlevé property
I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
5
votes
1
answer
519
views
Branched covers of the sphere branched over few points
Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...
- 6,548
2
votes
0
answers
154
views
Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
6
votes
1
answer
324
views
Almost complex manifold of dimension 2... locally isomorphic to ℂ?
I know that this is supposed to be standard, but I don't know how to search for it... hence the question:
Let $J$ be an almost complex structure on $M:=\mathbb R^2$, i.e., a $C^\infty$ section of $\...
- 43.2k
2
votes
0
answers
61
views
Criteria for a limit to be a proper function
This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
- 779
4
votes
0
answers
157
views
Modulus of an annulus with a cut
Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.
Let us ...
- 14.3k
2
votes
2
answers
417
views
Stoilow Theorem
I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A ...
- 85
4
votes
3
answers
687
views
Finite covers of punctured Riemann surfaces
Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$.
Let $Y_0$ be a smooth ...
- 21.8k
5
votes
1
answer
463
views
Structure of the automorphism group of a Riemann surface
I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...
- 2,696
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
7
votes
4
answers
3k
views
Classification compact Riemann Surfaces
I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?
In other words, is the complex structure ...
- 251
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 ...
6
votes
4
answers
2k
views
Space of $(1,0)$-holomorphic forms on a Riemann surface
In a complex analysis course I have been given the following definition:
Let $X$ be a Riemann surface, denote by $H(1,0)$ the space of all $(1,0)$-holomorphic forms on $X$ and consider the quotient ...
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