1
vote
1answer
107 views

About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps: If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$: ...
3
votes
1answer
102 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 ...
6
votes
2answers
183 views

Comparing different layered structures for fibered 3-manifolds: example request.

Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...
2
votes
1answer
195 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 ...
0
votes
0answers
139 views

Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...
0
votes
0answers
153 views

Quick references/sources for the hyperbolic Riemann Surfaces with boundary

Hello, Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...
0
votes
1answer
156 views

Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$

Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...
6
votes
1answer
249 views

For which surfaces is Penner's conjecture known to be true?

Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...
5
votes
2answers
810 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
vote
1answer
275 views

teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...
4
votes
1answer
332 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 ...
2
votes
1answer
365 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 ...
2
votes
1answer
649 views

How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions : A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...
5
votes
2answers
269 views

Optimal pants decompositions of a hyperbolic surface

Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$. Here is a candidate ...
3
votes
1answer
387 views

Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ...
1
vote
0answers
200 views

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello, This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states : If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
1
vote
1answer
329 views

Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states : Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...
4
votes
1answer
551 views

A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

Hi, On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves : Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U ...
3
votes
1answer
322 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 ...
0
votes
2answers
645 views

A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

Hello, I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
1
vote
1answer
437 views

Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello , I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I ...