All Questions
8 questions with no upvoted or accepted answers
6
votes
0
answers
209
views
Stable norm on hyperbolic surfaces
For a hyperbolic surface $S$ and a homology class $h\in H_1(S)$ its stable norm is defined as $\lim_{n\to\infty}\frac{1}{n}l(nh)$, where $l(nh)$ means the minimal length among all closed geodesics ...
- 10.4k
6
votes
0
answers
389
views
A conjecture of Thurston and possibly Weeks too
What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
- 1,131
3
votes
0
answers
120
views
Understanding $\kappa$-cones
I recently came across the concept of a $\kappa$-cones of a metric space (Chapter I.5.2) of Bridson and Haefliger's book. In their Proposition 5.8, the provide some intuition of $\kappa$-cones by ...
- 169
3
votes
0
answers
198
views
Uniform continuity of length function on geodesic currents
I'm starting to study geodesic currents and I have a question concerning uniform continuity.
Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...
- 31
2
votes
0
answers
177
views
Structure of hyperbolic manifolds of finite volume
Let $X$ be a hyperbolic manifold of finite volume.
I want to prove that $X$ has ends of the form $N\times \mathbb{R}$ where $N$ has a finite covering by a nilmanifold and $\pi_1N\to \pi_1 X$ is ...
- 563
1
vote
0
answers
161
views
Doubly ruled surfaces in hyperbolic 3-space
A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...
- 2,337
1
vote
0
answers
145
views
Comparison theorem for Lambert quadrilateral
A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle.
If $AOBF$ is a Lambert ...
- 319
0
votes
0
answers
88
views
Bound on the distance from points to the boundary of a hyperbolic surface
Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...
- 81