All Questions
Tagged with moduli-spaces teichmuller-theory
22 questions
0
votes
1
answer
159
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 "...
- 7,127
1
vote
0
answers
73
views
Local Chart for Teichmuller Space as A Manifold
Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem,...
- 51
3
votes
0
answers
104
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Relating different parametrizations of moduli space of Riemann surfaces
I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related:
On the one hand, there is a parametrization coming from hyperbolic ...
- 1,435
4
votes
0
answers
109
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Holomorphic maps on moduli space and Deformation theory
Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map
$$f:\mathcal{M}\rightarrow \mathcal{F}$$
means that for each ...
- 4,408
5
votes
1
answer
248
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Explicit check of the invariance of the Weil-Petersson form
Using Fenchel-Nielsen coordinates, the Weil-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition of ...
- 1,435
3
votes
1
answer
853
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Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures
In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
- 1,435
1
vote
1
answer
204
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Lie bracket on the complex valued functions of the space of representations of a Riemann surface
Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman (Invariant functions on Lie groups and Hamiltonian flows of surface group representations) proved that, for a fairly general class ...
- 323
11
votes
3
answers
953
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What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?
There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic ...
- 4,164
5
votes
1
answer
204
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Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...
- 989
3
votes
1
answer
797
views
Hyperbolic Metric on a Riemann Surface
From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The ...
- 989
2
votes
0
answers
260
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Parametrizations of the Moduli Space of Riemann Surfaces
I am looking for a reference or references about different parameterizations of moduli space of Riemann surfaces of genus $g$ with $n$ borders and/or punctures. I wish to know the basics of different ...
- 989
1
vote
0
answers
354
views
Generalized McShane Identity for Closed Riemann Surfaces
There is an identity for the hyperbolic Riemann surfaces with at least one border. The identity is known as Generalized McShane Identity or Mirzakhani-McShane Identity proved by Mirzakhani in her ...
- 989
5
votes
0
answers
306
views
Does the Torelli space appear "in nature"?
What I mean by the (slightly facetious) title is:
The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of ...
- 1,981
2
votes
0
answers
462
views
Teichmuller Space of a Disk with Holes and Boundary Punctures
If we consider a disk $D$ with $h$ holes and $n$ punctures on the boundary of the disk, then:
Is there a uniformization theorem for such surfaces?
What is the condition on $h$ and $n$ such that we ...
- 989
6
votes
2
answers
495
views
Riemann Theta Function On Hyperbolic Riemann Surfaces
The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by
$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...
- 989
7
votes
2
answers
1k
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Uniformizations of the bordered/punctured Riemann surfaces
The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different ...
- 989
8
votes
1
answer
388
views
Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?
It is known that Teichmüller distance ($d_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d_{WP}$) is not complete.
See for example the article
Wolpert, Scott. ...
- 227
7
votes
1
answer
389
views
Selberg Zeta Function and Fenchel-Nielsen Coordinates
According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...
- 989
2
votes
0
answers
322
views
Dessin d'enfant and moduli space of bordered/punctured hyperbolic Riemann surfaces
Belyi's theorem states that if a Riemann surface could be defined as an algebraic curve over an algebraic number field, then this Riemann surface could be described by a Dessin d'enfant. I have two ...
- 989
9
votes
1
answer
858
views
A question about Mirzakhani et. al.'s algebraicity theorem
While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...
- 13.8k
42
votes
1
answer
20k
views
What is "Teichmüller Theory" and its history?
What is "Teichmüller Theory"? What part has been worked out / foreseen by O. Teichmüller himself and what is further development? Is there some current work which might be considered as continuation/...
- 24.9k
3
votes
1
answer
899
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Basic Questions about Teichmuller's theorem/quadratic differentials
I have some basic questions about Teichmuller's theorem, since I am a beginner, my questions might be very basic. If you can give some hints/answers or cite some references to study from, I will ...
- 1,471