Consider a planar pair of pants 
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r\_1, |z+x| \ge r\_2 \right\rbrace$$
where $-1 < -x-r\_2 < -x+r\_2 < 0 < x-r\_1 < x+r\_1 < 1$.


There is a unique conformal hyperbolic metric $e^{2u} (d x^2 + d y^2)$ on $P$ such that all boundary components are geodesics. This corresponds to the unique solution $u:P \to \mathbb{R}$ to $\Delta u = e^{2u}$ with constant normal derivative equal to the inverse of the radius (with a minus sign on the outer boundary and a plus sign on the two others) on each boundary component.


The lengths of the boundary components for this unique metric is a complete conformal invariant for $P$. By this I mean that two pants are conformally equivalent if and only if the three lengths are equal.


**My questions:**


1. Given a pair of pants can one estimate what the three lengths of the boundary geodesics will be?
2. Are there known relationships to other conformal invariants such as modules of rings, etc?
3. Do you know of any reference where this particular conformal invariant is considered/studied?


So far I've made some progress on question 1. By considering different anulii and using the definition of extremal length I can bound certain hyperbolic distances from above (be it the length of a boundary curve, or the distance between two of them). Using the Collar Lemma this also yields lower bounds.


However I'm guessing something better is known (?).



Conformal invariants of planar pairs of pants
There are several references that consider the relation between extremal length and hyperbolic length. Usually they consider closed surfaces, but you can put yourself in that situation by doubling along the boundary. Here are some relevant papers:


Matsuzaki, Katsuhiko. Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces. Geometric complex analysis (Hayama, 1995), 443–450, World Sci. Publ., River Edge, NJ, 1996. <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.7078>


Maskit, Bernard. Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386.


Rafi, Kasra. Thick-thin decomposition for quadratic differentials. Math. Res. Lett. 14 (2007), no. 2, 333–341. <http://www.math.toronto.edu/~rafi/Papers/Thick-Thin.pdf>


I think the Maskit paper might be most relevant.


