All Questions
5 questions
13
votes
2
answers
484
views
How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
3
votes
2
answers
1k
views
Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
6
votes
2
answers
495
views
Riemann Theta Function On Hyperbolic Riemann Surfaces
The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by
$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...
1
vote
0
answers
245
views
Schwarz–Ahlfors–Pick theorem for hyperbolic pair of pants
Let $\mathcal{P}$ be a hyperbolic pair of pants with geodesic boundary and $\Sigma_3$ be the hyperbolic thrice punctures sphere. I want to construct a conformal map $f:\mathcal{P}\rightarrow \Sigma_3$ ...
3
votes
2
answers
618
views
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...