Questions tagged [teichmueller-theory]
Questions related to the work and continuation of Oswald Teichmüller on Teichmüller theory, especially Teichmüller spaces.
111 questions
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Orientation for groups
I am significantly rewriting this question, as, I hope, I can formulate it now better from a different angle.
Is there such a concept as an orientation for a group? (For some classes of groups?)
Here ...
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Why is bers' Beltrami differential harmonic?
I am reading the book "An introduction to Teihmuller spaces" by Y. Imayoshi and M. Taniguchi. In chapter 6, p153, when constructing a local inverse of Bers' Embedding, Bers' Beltrami ...
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Questions on Relative Forms and the de Rham Theorem in Geometry of Algebraic Curves, Vol. II
In Geometry of Algebraic Curves, Vol. II, by Arbarello, Cornalba, and Griffiths, they use the notation $\mathcal{A}^q_{X/Y}$ to define the sheaf of smooth relative $q$-forms for a smooth fibration $X \...
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Induced flat metric from holomorphic quadratic differential
Given a holomorphic quadratic differential $q$ on a Riemann surface, there is a natural coordinate to make $q=z^kdz\otimes dz$ locally.
On A Primer on Mapping Class Groups, p.312, it says, "The ...
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Explicit description of Fuchsian group for hyperbolic surfaces of finite type
Let $S_g$ be a Riemann surface with genus $g>1$. Uniformization theorem says the universal covering of $S_g$ is the upper half plan $\mathbb{H}$ and $S_g$ is conformal equivalent to $\mathbb{H}/\...
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Renormalization in holomorphic dynamics
Renormalization seems to play an ever increasing in scope and importance role in the field of holomorphic dynamics, but there seem to be hardly any modern resources adressing the principal ideas ...
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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 ...
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Existence of critical trajectories of quadratic differential
Let $P$ be a polynomial with complex coefficients. A horizontal trajectory is a curve in $\mathbb{C}$ such that $P(z) \, dz^2 > 0$, i.e. a solution $\gamma$ to the differential equation $$P(\gamma(...
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Split and differentiable locally trivial submersion, and more confusions on a proof in John H Hubbard's book
The "split" means
Let E, F be Banach spaces, A surjective linear map $f:E\rightarrow F$ is said to split if $ker f$ admits a closed complement. More generally, a submersion is a split ...
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A question about Möbius transformation and Beltrami forms
In the Page 170, Proposition 4.8.19 of John H. Hubbard's book "Teichmuller theory and applications to geometry, topology and dynamics", the author said:
Let $A: P^{1}\rightarrow P^{1}$ be a ...
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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 ...
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Existence of extremal map in teichmuller class
Let $R$ be a Riemann surface, and define two quasiconformal maps $f_i : R \rightarrow R_i, i=1,2$ to be equivalent if there exists a conformal map $c : R_0 \rightarrow R_1$ and a homotopy $g_t : R \...
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Holomorphic 1-forms as a subspace of the first de Rham cohomology group of a surface
If we fix a complex structure $c$ on a closed oriented surface $S$, then the space of holomorphic 1-forms $\Omega^1(S,c)$ has complex dimension equal to the genus $g$ of $S$ and - since they are all ...
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Volume of strata of abelian differentials
I am reading this paper https://arxiv.org/pdf/math/0006171.pdf on the volume of strata in the moduli space of abelian differentials, and just have a small clarificatory question.
On page 4, the volume ...
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Non-compact Riemann surfaces with zero Euler characteristics
The question arises because of a statement in John H.Hubbard's book "Teichmuller theory and Applications to Geometry, Topology and Dynamics, vol 1" Page 130, Chapter 4.4.It says that:
The ...
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Understanding a curve system on the boundary of Teichmuller space
I am reading the paper:
https://arxiv.org/pdf/1511.07635.pdf
and I try to understand the behavior on the boundary. I have some problems understanding this definition.
On page 26, the authors give a ...
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Are the Teichmuller spaces equivalent?
I have this question in my mind since a long time. But I am failed to find an answer.
The following figures are closed, connected, orientable surfaces of genus $2$.
Topologically both surfaces are ...
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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 ...
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$\Gamma_{\tilde{P}}$ is a free group generated by two hyperbolic transformations
Book: An introduction to Teichmuller spaces by Imayoshi & Taniguchi.
Let $R$ be a Riemann surface whose universal cover is $\Bbb D^2$ (i.e. Hyperbolic surface). Consider cutting $R$ by a family of ...
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Pure mapping classes
I am a new guy in mapping class groups and I am very confused about pure mapping classes tonight. From the Primer by Farb and Margalit, I learned the definition of pure mapping class groups PMod$(S_{g,...
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Isometric embedding of Teichmüller spaces
Let $S_{g,k}$ be a genus $g$ surface with $k$ punctures. Let $\mathrm{Mod}(S)$ be the extended mapping class group of a surface, defined as the isotopy classes of the self-homeomorphisms of surface $S$...
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Constructing a pseudo-Anosov homeomorphism
Let S be a genus 2 closed surface. Consider the two non-separating red curves (say $\alpha$ & $\beta$) in the picture. I want to construct a pseudo-Anosov homeomorphism $\phi: S \rightarrow S$ ...
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Convergence in Teichmüller space
Question 1
Let $X$ be a compact hyperbolic Riemann surface and let $\{[\mu_n]\}$ be a sequence of points in the Teichmüller space $\mathcal T(X)$.
Further, assume there are maps $f_n:X \xrightarrow{q....
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Does moving a small enough distance in Teichmüller space change the marking?
Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \...
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Why is the Thurston pullback map well-defined?
If $f:S^2 \rightarrow S^2$ is a Thurston mapping of degree $d$ (a finite ramified covering map with finite postcritical set $P_f$).
The Thurston pullback map $\sigma_f: \mathcal{T}_{S^2 \setminus P_f} ...
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Reference request: bundle of holomorphic differentials over Teichmüller space
Let $S$ be a compact orientable surface of genus $\geq 2$ and let $\mathcal{M}_{-1}(S)$ be the space of smooth hyperbolic metrics on $S$. Then one can define the Teichmüller space on $S$ as $\mathcal{...
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Different definitions for the Teichmüller space of puctured spheres
My question is about understanding of equivalence of two different definition for the Teichmüller space of hyperbolic surfaces $\mathbb{S}^2 \setminus P$, where $|P| \geq 3$.
The first definition for ...
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Question about Hubbard's analytic definition of quasiconformality. Aren't weak derivatives only defined up to a set of measure zero?
I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/...
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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 ...
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When is a g-dimensional subspace of $H^1(S; \mathbb{C})$ the $H^{1,0}$ of a complex structure on $S$?
Given a closed surface $S$ of genus $g \geq 1$ there are lots of choices of complex structure, and each one singles out a subspace of the surface's first cohomology group with complex coefficients ...
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Automorphism classes of branched covers of disks over disks
I was reading Hubbard's book (Teichmuller Theory Vol 2) and in a proof (9.3.2), he mentions that there is one branched cover of the disk over a disk with 1 ramification point (degree $k$) up to ...
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Learning Roadmap for 3-dimensional Hyperbolic Geometry
I am interested in learning Teichmuller theory. Previously, I learnt 2 dimensional hyperbolic geometry from S. Katok's book. Now, I am interested in learning about significance of hyperbolic geometry ...
user886636
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Quotient of hyperbolic disc model
I am trying to learn decorated Teichmüller theory and reading Penner's "Decorated Teichmüller Theory" book. In Chapter 2 in "Punctured Surfaces" section I got confused.
Let $\...
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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 "...
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References on hyperbolic geometry and Teichmuller Theory
I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
user886636
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Lemma 3.6 in Kerckhoff's "The Nielsen Realization Problem"
I am confused by the proof of Lemma 3.6 in Kerckhoff's "The Nielsen Realization Problem." Let $\gamma \in S$ ($S$ is the set of isotopy classes of simple closed curves) and $\bar{\gamma}(t)$ ...
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Thurston Compactification
Sorry in advance for my English.
I'm studying the Thurston compactification from the Jean-Pierre Otal's book "The Hyperbolization Theorem for Fibered 3-Manifolds".
I have a question, what $\mathbb{...
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When might Fenchel-Nielsen twist coordinates exceed 1/4?
When a compact Riemann surface of genus $g$ is cut up along $3g-3$ disjoint geodesic loops into $2g-2$ pairs of pants, the result is often described by giving Fenchel-Nielsen coordinates: one length ...
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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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Fourier coefficients of a function
Prove that for $w(\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{z}-1)(\...
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Induced map from the universal cover of the base space to the Teichmuller space of the fiber is holomorphic
Let $\phi:X\to Y$ be a submersion and holomorphic map with not every two fibers are biholomorphic, where $X$ a compact complex surface, $Y$ is a Riemann surface. Let $\pi:U\to Y$ be the universal ...
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how do the Lp spaces sit inside Teichmüller space?
I've heard it said that Teichmüller space gives a metric to the space of all metric spaces.
If this is so, where do the Lp spaces sit in the space of all metric spaces?
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Homeomorphism Between Closed Riemann Surfaces Homotopic to Quasiconformal Mapping
I'm re-reading a paper of Bers and for the second time, and I am yet again confused about the claim in the title, which Bers declares to be easy to prove.
For context, I'll lay out some terminology....
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Atlas for hyperbolic pair of pants
I was reading about Teichmüller spaces and how you give a hyperbolic structure to a pair of pants via glueing two right angled hexagons, but I wanted to know if there was an explicit way to describe ...
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A Third Derivative in Ahlfors' `Some Remarks on Teichmuller Space'
I'm slugging my way through Ahlfors' "Some Remarks on Teichmuller Space" and am stuck on the calculation of equation (1.18). The problem here is to compute the third derivative, on the unit disk $\...
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Surjectivity of parameters for torus double cover of sphere
Let $T=\mathbb{C}/\Lambda$ be a torus, viewed as a Riemann surface.
Quotienting out by the relation $z\sim -z$ gives a double cover $\pi : T \to \hat{\mathbb{C}}$ of the Riemann sphere, ramified over ...
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Understanding Teichmuller equivalence of marked compact Riemann surfaces
Let me give some definitions to begin with.
Let $\Lambda$ be a compact orientable two-dimensional (over $\mathbb R$) manifold of genus $p>2$. Let $g$ be a conformal structure on $\Lambda$, and $...
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Could someone explain to me what a Teichmuller Space is?
In the simplest terms possible, for someone who understands the basics of Manifolds, Topology, but barely any of the more complicated topics.
I've been using the following:
http://homeowmorphism.com/...
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Moduli Space of Tori identified with $\mathbb{C}$
I am currently reading through the book An Introduction to Teichmüller Spaces by Imayoshi and Taniguchi. In Section 1.2, we see that $M_1$, the moduli space of tori, can be identified with $\mathbb{H}/...
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Why is the matrix representation of an almost complex structure like this?
Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
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