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Results tagged with riemann-surfaces
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user 39654
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
4
votes
Comparison of special metrics on Riemann Surfaces with the hyperbolic one
First of all, the constants $c_i$ will have to depend on the complex structure of $X$ since without prescribing a complex structure one cannot talks about dependence on a basis of the space of holomor …
- 12.3k
0
votes
Accepted
Thurston's metric is bounded
I am not sure how well you know the Teichmuller theory, but the basic thing to understand is that $\mathcal T$ is not the space of hyperbolic metrics on $\Sigma$: It is the space of pairs $(\sigma, [\ …
- 12.3k
5
votes
Accepted
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Expanding on my comments, here are some obstructions coming from Hausdorff dimension and self-similarity.
One observation is that every nonelementary Kleinian group $\Gamma$ has positive critical exp …
- 12.3k
3
votes
Accepted
Can we strengthen this exercise in Forster's book on Riemann surfaces?
The answer to your first question is positive. First of all, it suffices to consider the case when every point of your subset $C\subset S$ is a non-removable singularity of $f: S\to \mathbb C P^1$. (I …
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5
votes
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Lengths of generators of surface group
In order to remove this question from the "unanswered list." Let $\epsilon>0$ be the Margulis constant for the hyperbolic plane (with curvature $-1$). Then for every complete hyperbolic surface $S$, i …
- 12.3k
22
votes
When does a group act effectively and holomorphically on some Riemann surface?
Donu's answer is correct but amounts to killing a fly with a gun shot: Greenberg proves a harder result than the one needed for the problem.
Theorem. Let $G$ be a countable group. Then there exists a …
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5
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Holomorphic maps from a Riemann surface of infinite genus
The following is not a real answer but an extensive comment on the OP.
First, recall that an (open) Riemann surface is said to have type $P_{AB}$ (resp. $O_{AB}$) if it admits (resp., does not admit) …
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13
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Can you cover a genus a billion hyperbolic surface with 15 balls?
Your conjecture is false. Every nonorientable closed connected surface of negative Euler characteristic, admits a hyperbolic metric such that the surface is covered by 3 embedded disks. Hence, for eac …
- 12.3k
4
votes
Accepted
Positive genus Fuchsian groups
Yes, this is true, but proving this is easier than finding a reference.
Every finitely-generated matrix group (e.g. a lattice in $PSL(2, {\mathbb R})$ contains a torsion-free subgroup. The general re …
- 12.3k
8
votes
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
Here is a translation of Strebel's definition.
First of all, a quadratic differential on a complex manifold $M$ (holomorphic or not) is a smooth section of the symmetric square $S^2(T^{*(1,0)}M)$ of …
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