1
$\begingroup$

I am trying to understand the article by Maryam Mirzakhani about Simple geodesics and Weil-Petersson volumes. In the third section of this article, the following proposition is stated. And I want to know why this proposition is correct. I mention the proposition:

Proposition: Let $P$ be a hyperbolic pair of pants with (non-empty)geodesic boundary components $\beta_1 , \beta_2, \beta_3$ of lenghts $x_1, x_2, x_3$ respectively. Then $P$ contains $5$ complete geodesics disjoint from $\beta_2 , \beta_3$ and orthogonal to $\beta_1$. More precisley, two of these geodesics meet $\beta_1$ respectively at $y_1,y_2$ and spiral to $\beta_3$, the other two meet $\beta_1$ respectively at $z_1,z_2$ and spiral to $\beta_2$. there is also a unique common geodesic perpendicular from $\beta_1$ to itself meeting $\beta_1$ perpendicularly at two points $w_1, w_2$.

enter image description here

$\endgroup$
  • $\begingroup$ I don't have time right now to draw pictures, but: To see the unique geodesic from $\beta_1$ to itself, think of the pair of pants as a pair of right-angled hexagon, the saddle geodesic will be apparent. To see the spiraling geodesics, develop the pair of pants into $\mathbb{H}^2$, then you will be able to find a geodesic perpendicular to the developing image of $\beta_1$ and asymptotically approaches the developing image of $\beta_3$. This is in 3.9 of Thurston's notes. $\endgroup$ – Neal Sep 16 at 15:25
  • 1
    $\begingroup$ Actually, none of these is a complete geodesic, rather, it is a geodesic ray $\endgroup$ – Moishe Kohan Sep 19 at 14:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.