All Questions
9 questions
11
votes
3
answers
748
views
Explicit triples of isomorphic Riemann surfaces
Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....
4
votes
2
answers
439
views
Simple Closed Hyperbolic Geodesics on Punctured Spheres
Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
- 1,761
2
votes
1
answer
111
views
Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
- 257
2
votes
2
answers
290
views
Subharmonic function on a twice punctured complex plane
is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?
Thanks,
4
votes
1
answer
503
views
Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains
Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we ...
- 1,471
3
votes
1
answer
907
views
The version of Montel's theorem used in the proof of Jenkins-Strebel differential
Hello,
I am afraid that my main question might be a bit too elementary, but still I ask :
In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
- 1,471
2
votes
2
answers
650
views
Hurwitz's automorphisms theorem for infinite genus Riemann surfaces
Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...
- 1,159
4
votes
6
answers
925
views
Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$
Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...
- 1,471
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 ...
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