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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....
Joseph O'Rourke's user avatar
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 ...
Willemien's user avatar
  • 305
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/...
Roice Nelson's user avatar
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 $i(\bar{a})$ ...
Cusp's user avatar
  • 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 ...
Cusp's user avatar
  • 1,713
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 $...
Mario's user avatar
  • 41
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 ...
Cusp's user avatar
  • 1,713
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-...
Cusp's user avatar
  • 1,713
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?
Gus's user avatar
  • 85
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 ...
user158773's user avatar
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 ...
Cusp's user avatar
  • 1,713
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 ...
giulio bullsaver's user avatar
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 ...
Cusp's user avatar
  • 1,713
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 ...
Cusp's user avatar
  • 1,713
0 votes
0 answers
89 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-...
AW.'s user avatar
  • 21
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\...
Roberto Frigerio's user avatar