All Questions
Tagged with teichmueller-theory moduli-space
11 questions
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Understanding Gauss-Manin connection on Arbarello's book
I am trying to understand the Gauss-Manin conecction in order to understand the definition of the Period Mapping on the Moduli space of algebraic curves of genus $g$ and its extension to the ...
- 935
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Moduli spaces of surfaces to algebraic stacks
I've been reading through Farb and Margalit's book on the action of the mapping class group on Teichmuller space to get a moduli space. This is a very topological/geometric construction, looking at ...
4
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What is Higher Teichmuller Theory?
I am interested in representation of surface group. So, I am studying the book Lectures on Representations of Surface Groups by Francois Labourie. The main goal of representation of surface groups is ...
- 47
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449
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Teichmuller space of the 4-punctured sphere
I'm a bit confused working with Teichmuller space at the moment. Let's think of Teichmuller space as the space of holomorphic/conformal structures on a surface, up to diffeomorphisms isotopic to the ...
- 2,378
7
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831
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What is Representation of Surface Groups?
I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
- 93
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82
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Singularities of moduli spaces $M_g$
This survey paper from Lizhen Ji says:
Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he
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- 3,726
2
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1
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123
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Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
Assume that $\mathcal{R}$ is a hyperbolic surface with $m$ geodesic boundary components and $n$ punctures. If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is ...
- 253
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2k
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Hyperbolic Metric on a Riemann Surface
From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the ...
- 253
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302
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Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations
The theorem of existence proved by Kodaira and Spencer shows that a complex manifold $M$ for which the second cohomology group $H^2(M, \Theta T^{1,0} M)$ is trivial, $\Theta T^{1,0} M$ being the sheaf ...
- 1,357
2
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1
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719
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Dehn twist as isometries on hyperbolic surface
[I am editing the question to a most correct and precise one thanks to comments of Lor and studiosus]
Let (S,g) be a compact hyperbolic surface. On a simple closed geodesic $\gamma
$ I can realized a ...
- 43
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331
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Hyperbolic (and related) structures on open unit disk
I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
- 5,628