All Questions
Tagged with hyperbolic-geometry curves-and-surfaces
16 questions
0
votes
0
answers
88
views
Existence of a geodesic on a non-orientable surface
Let $\Sigma$ be a non-orientable surface possibly with boundary or punctures. Is it possible that a one-sided loop in $\Sigma$ is always realized as a geodesic?
In the orientable case, it is well-...
- 21
17
votes
2
answers
2k
views
Geodesics on the twisted pseudosphere (Dini's surface)
I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p....
- 326
4
votes
3
answers
2k
views
Two curves filling a surface
Let $S$ be a closed surface of genus $g \geq 2$. Do there exist two simple closed curves filling $S$?
Definitions:
Two closed curves $\alpha$, $\beta$ fill $S$ if they have minimal intersection and $...
- 12.9k
2
votes
1
answer
213
views
Hyperbolic length of curve that does not enter a collar
Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
- 28.2k
2
votes
1
answer
152
views
Coordinates for Laminations: geometric versus shear
Let $S$ be an orientable surface with a triangulation T.
A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...
- 28.2k
10
votes
2
answers
1k
views
Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?
There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
- 63.9k
3
votes
1
answer
155
views
Intersecting geodesics on a surface from non-intersecting geodesics
Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
- 28.2k
1
vote
0
answers
143
views
Change of length of curve when Fenchel-Nielsen length coordinate increase
Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
- 1,713
7
votes
3
answers
414
views
In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?
Given a Jordan curve on the $\mathbb{H}^3$ boundary at infinity, there is a minimal surface (topological disk) in $\mathbb{H}^3$ with the curve as its asymptotic boundary (page.mi.fu-berlin.de/...
- 68.9k
1
vote
0
answers
126
views
Is triple point intersection 'generic' in Teichmuller space?
Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
- 19.6k
2
votes
1
answer
113
views
Is the length function associated with the twist parameter an increasing function?
Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is ...
- 96.4k
6
votes
1
answer
317
views
Reduction of self-intersections without reducing the geometric intersection
Let $F$ be a hyperbolic surface. Given a closed curve $a$, let $\bar{a}$ denotes the free homotopy class of $a$. Let $i(\bar{a},\bar{b})$ denotes the geometric intersection number and $i(\bar{a})$ ...
- 68.9k
4
votes
1
answer
814
views
Angle between geodesics in hyperbolic surface
Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...
- 15.4k
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
5
votes
1
answer
193
views
Injective simplicial maps between Arc complexes
Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
- 15.5k
0
votes
0
answers
177
views
Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation $\rho\...
- 3,749