All Questions
Tagged with complex-geometry riemann-surfaces
122
questions
2
votes
1
answer
121
views
Branched covering maps between Riemann surfaces
What is an example of a branched covering map between Riemann surfaces of infinite degree? i.e. something like a branched version of the exponential map $exp: \mathbb{C} \to \mathbb{C}^*$.
Thanks!
- 357
6
votes
0
answers
200
views
Holomorphic fibre bundles over noncompact Riemann surfaces
Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...
- 271
0
votes
1
answer
146
views
Teichmüller theory for open surfaces?
I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...
- 6,996
4
votes
1
answer
255
views
Reference for Teichmuller spaces of punctured surfaces
What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$?
I am looking for a reference where there is the correct statement and or proof of say the Bers embedding,...
- 617
3
votes
1
answer
253
views
Hyperelliptic integrals
I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
- 89
2
votes
1
answer
217
views
Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
- 77
4
votes
2
answers
385
views
Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question ...
- 452
4
votes
2
answers
345
views
Holomorphic Gauss normal map
Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$.
Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic ...
- 452
1
vote
0
answers
214
views
Unexpected holomorphic tubular neighborhood
While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
- 1,731
6
votes
0
answers
278
views
What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?
$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...
- 565
2
votes
0
answers
289
views
Uniformization of Riemann surfaces with cone singularities
Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such ...
- 254
8
votes
2
answers
391
views
Holomorphic maps from a Riemann surface of infinite genus
Let $X$ be a Riemann surface of infinite genus and let $n$ be an arbitrarily large natural number.
Do there always exist a closed Riemann surface $Y$ of genus greater than $n$ and a nonconstant ...
- 81
3
votes
1
answer
143
views
Existence of covering isomorphism
Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...
- 73
7
votes
1
answer
271
views
Riemann uniformization theorem (limit case)
Let $\mathbb D_r=\{z\in\mathbb C:|z|\le r\}$ be the closed unit disk of radius $r$,
let $\mathring {\mathbb D}_r=\{z\in\mathbb C:|z|< r\}$ be its interior,
and let $\mathbb A_r=\mathbb D_r\setminus ...
- 42.7k
-1
votes
1
answer
92
views
Related to the Schwarz Christoffel map
With the help of the Schwarz-Christoffel map, for a given polygon (given angle), we can find some points on the boundary of the upper half plane, such that a particular Schwarz-Christoffel map takes ...
- 541