All Questions
Tagged with hyperbolic-geometry riemann-surfaces
83 questions
26
votes
4
answers
1k
views
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
- 261
20
votes
3
answers
1k
views
Failure of Mostow rigidity in dimension 2
I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question:
(1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...
- 11.9k
20
votes
5
answers
3k
views
Finding Constant Curvature Metrics on Surfaces without full power of Uniformization
(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)
Let S be a surface (possibly non-compact, but no boundary). It seems that there are ...
- 3,203
20
votes
1
answer
1k
views
Canonical immersion of the double torus
It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
- 2,091
18
votes
2
answers
1k
views
The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions
I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
- 385
17
votes
1
answer
2k
views
How can I calculate the period matrix of this Riemann surface?
I am attempting to calculate the period matrix of the Riemann surface associated to the zero set of a complex curve: $y^3 = x^4 -1$.
Background:
It is my understanding that the period matrix of a ...
- 3,539
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
14
votes
1
answer
852
views
Cutting up the Bring surface into six pairs of pants
The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....
- 423
13
votes
2
answers
485
views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
- 10.1k
13
votes
2
answers
484
views
How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
- 5,063
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....
11
votes
2
answers
811
views
Can you cover a genus a billion hyperbolic surface with 15 balls?
Here's a question I was wondering about this week. Not sure how interesting it is, but I thought it was kind of curious.
Question: Given $k$, is there a number $N=N(k)$ such that if a closed ...
- 532
11
votes
0
answers
278
views
Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
- 1,435
10
votes
3
answers
2k
views
Hyperbolicity on Riemann Surfaces
For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a ...
- 3,626
10
votes
1
answer
320
views
Systole of Riemann surfaces of genus $g$
In Buser and Sarnak's "On the period matrix of a Riemann surface
of large genus", we get
$$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
- 181
9
votes
2
answers
2k
views
Translation surfaces
I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete ...
- 1,471
9
votes
2
answers
3k
views
What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?
I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with ...
- 1,471
8
votes
2
answers
826
views
Weil's theorem about maps from a discrete group to a Lie group.
Let K be a group (with discrete topology), G be a Lie group. Let $\operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps ...
- 3,203
7
votes
1
answer
700
views
Bers' constant for compact hyperbolic surfaces with geodesic boundary
The clasical Bers' theorem about pants decomposition says that any compact Riemann surface of genus $g \geq 2$ has a pants decomposition such that every cutting geodesic in this decomposition is of ...
- 305
7
votes
2
answers
1k
views
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
7
votes
0
answers
218
views
Purely analytic proof of the Nielsen-Thurston classification theorem
I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...
- 191
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
5
votes
4
answers
1k
views
Softwares for drawing hyperbolic surfaces , closed, with boundaries or with punctures ?
In a paper I am in the process of writing in LaTeX, I need to draw and incorporate some diagrams of hyperbolic surfaces in my LaTeX document. Is there any software I can use to draw hyperbolic ...
- 1,471
5
votes
1
answer
722
views
Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps
If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get ...
- 1,471
5
votes
1
answer
104
views
When do two measured foliations on a surface define a Riemann surface structure?
Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
- 275
5
votes
1
answer
142
views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
- 14.3k
5
votes
1
answer
879
views
Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues
Hello,
Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-...
- 1,471
5
votes
1
answer
447
views
Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$
I have a reference request for a proof for the following statement in the title:
The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
- 153
5
votes
2
answers
495
views
Fenchel-Nielsen length-length coordinates on Teichmueller space?
Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...
- 2,274
5
votes
1
answer
204
views
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
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
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
4
votes
1
answer
772
views
Visualizing hyperbolic metric of punctured sphere
Uniformization of the 3-punctured sphere generates a "pants" configuration with three legs narrowing down to cusps. This is supposed to have a metric of constant negative curvature, and I can see this ...
- 101
4
votes
1
answer
384
views
hyperbolic orbifolds of small area
Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller ...
- 16.6k
4
votes
1
answer
1k
views
Hyperbolic structures on once punctured tori
I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...
- 1,601
4
votes
1
answer
213
views
Conformal map between flat and hyperbolic torus with a boundary
I am confused because I can define two very different complex structures on the torus with a puncture/boundary.
For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
- 355
4
votes
2
answers
261
views
Measured geodesic laminations have either discrete or Cantor set local cross-sections
I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...
- 765
4
votes
1
answer
512
views
fundamental domains in H^2 containing large balls
I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
- 161
4
votes
1
answer
505
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
4
votes
1
answer
470
views
Cluster algebras of type A and X
I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces.
Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
- 1,435
3
votes
2
answers
1k
views
Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
user147962
3
votes
1
answer
886
views
The smallest positive eigenvalue and the length of the shortest geodesic
I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ ...
- 9,497
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 ...
user2529
3
votes
1
answer
853
views
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
3
votes
1
answer
563
views
Intersection of closed geodesics in hyperbolic surface
This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed non-...
- 1,713
3
votes
1
answer
177
views
Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
- 254
3
votes
1
answer
599
views
What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
- 1,435
3
votes
1
answer
432
views
Euclidean surfaces with conical singularities and cusped hyperbolic surfaces
Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$.
Construction 1: it is well-known that the conformal class $[g]$ ...
- 347
3
votes
1
answer
908
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
3
votes
1
answer
785
views
A question on part of "An introduction to teichmuller spaces" by Imayoshi-Taniguchi
I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say:
Since
$\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued ...
- 99