Questions tagged [riemann-surfaces]
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
688
questions
3
votes
1
answer
160
views
Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
- 254
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
5
votes
1
answer
83
views
When do two measured foliations on a surface define a Riemann surface structure?
Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
- 275
1
vote
0
answers
32
views
Simple smooth functions on Bolza surface
Consider the Bolza surface, a compact Riemann surface of genus 2.
It is an octagon in the Poincaire disk model with opposite sides identified.
I would like to write down some analytic expressions for ...
- 177
2
votes
1
answer
168
views
Reference request: uniformization theorem proof by Borel
This answer refers to a proof of the uniformization theorem via the PDE describing metrics of constant curvature (Liouville?) by Borel. I haven’t been able to find this reference, is anyone aware ...
- 631
1
vote
0
answers
47
views
Are Bergman metrics on compact Riemann surfaces continuous on Teichmüller space?
Let $R$ be a compact Riemann surface of genus $\geq 1$, and let $\omega_1,\ldots,\omega_g$ be holomorphic one forms that form the dual basis of canonical homology basis. Let $(\pi)_{ij}$ be the ...
2
votes
1
answer
200
views
Making a map in sheaf cohomology involving a theta characteristic explicit
Motivation:
For a given rank 2 vector bundle we want to know how many theta-characteristic valued twisted endomorphisms it has.
Setting:
Let $C$ be a smooth algebraic curve over a field of ...
9
votes
0
answers
126
views
Symplectic form on the space of geodesic currents on a surface?
There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...
- 67.8k
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
74
views
On nontrapping manifolds
Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary.
Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
- 4,113
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
0
votes
0
answers
105
views
Serre duality for non-compact Riemann surfaces
Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology
$$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$
Can we say ...
- 498
1
vote
0
answers
109
views
Describing the hyperbolic structure of punctured torus in terms of the period lattice
Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...
- 7,313
2
votes
1
answer
90
views
Coupling small and large injectivity radii
I'd like to know whether a manifold of constant curvature, which has large injectivity radius at many points, can have points of arbitrary small injectivity radius.
More precisely, for a point $x$ in ...
- 183