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:**

- Given a pair of pants can one estimate what the three lengths of the boundary geodesics will be?
- Are there known relationships to other conformal invariants such as modules of rings, etc?
- 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 (?).