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Results tagged with riemann-surfaces
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user 21684
Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
6
votes
Accepted
Special holomorphic triples on Riemann surfaces and branch divisors
See the formula in the proof of Corollary 11.3.1 in The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces:
$$
deg(L)= g-1 +(deg(E) -deg(B_s))/2.
$$
As for your first question (can we …
- 31.2k
5
votes
Accepted
Can the limit set of an infinitely generated Schottky group have positive area?
Two relevant references:
W. Abikoff, Some remarks on Kleinian groups. 1971 Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) pp. 1–5. Ann. of Math. Studies, No. 66. P …
- 31.2k
15
votes
Accepted
Does every Riemann surface with boundary immerse in C?
A more general result is proven in
Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).
As for compact surfaces with boundary, it is essentially a par …
- 31.2k
5
votes
Accepted
Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$
Let me start by interpreting the question "What kind of surface is $S$?" in the case of a general connected oriented topological surface (without boundary). (I am considering only oriented surfaces ju …
CommunityBot
- 1
6
votes
Accepted
Does every canonical decomposition of the intersection form come from a canonical homology b...
Yes, every such decomposition gives rise to a symplectic automorphism $h$ of $Z^{2n}$ (sending standard symplectic generators to the generators of $A$ and $B$ respectively).
Now, use the fact that th …
- 31.2k
1
vote
Accepted
A continuous version of Teichmuller uniqueness
Here are some details. First, in your setting, all the maps $g_n$ are K-quasiconfirmal for a certain K. Thus, by the convergence property for qc maps, the sequence $(g_n)$ has a convergent subsequence …
- 31.2k
12
votes
What prevents a cover to be Galois?
I assume that the surfaces are connected. Composition of two regular covering maps need not be regular even if the coverings are unbranched: A (usual) covering map is regular iff it is defined by a no …
- 31.2k
8
votes
Accepted
Fuchsian groups and their normalizers
Unfortunately, as far as I know, nobody really explains such things as they are considered to be "too elementary". The most basic, briefest and down-to-earth reference I know for the needed background …
- 31.2k
2
votes
Accepted
Automorphisms of higher-genus Riemann surfaces act nontrivially on homology (Reference Request)
The result itself seems to be due to A. Hurwitz:
"Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.
At least, this is what Babai refer to on page 42 …
- 31.2k
6
votes
Accepted
Complete metric on a Riemann surface with punctures
A good reference for this is, say, Kobayashi and Nomizu "Foundations of Differential Geometry". The result you are looking for is: If $M$ is a complete Riemannian manifold and
$p: M\to M'$ is a (loca …
- 31.2k
10
votes
Accepted
Embedding a Riemann surface in the sphere
See e.g. here:
Theorem 3.2.7. Any planar connected Riemann surface is biholomorphic to an open subset of $S^2$.
The proof is very straightforward: Exhaust a genus $0$ surface $S$ by relatively comp …
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