Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l\_1,l\_2,l\_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat representative of the form $S^1 \times [0,h]$, where we choose the boundaries to have some fixed length $2 \pi$.


What is $h$ as a function of $l\_1,l\_2,l\_3$?



What's the height of the capped hyperbolic pants?
EDIT - Maxime has answer the question in the comments. The desired $h$ is not a function of the lengths $\ell\_1, \ell\_2, \ell\_3$, but rather lies in an open interval. The minimum (resp. maximum) of $h$ is achieved by sewing the first circle shut so it lies along a concentric (resp. radial) arc of the resulting annulus. See Figure 46e in Nehari's book for a picture of the concentric case. 


I wrote the following under the assumption that there is a unique best way to attach a disk. I'll leave it here as it seems like a particularly nice intermediate case. 


%%%


It suffices to answer the following question. 


Fix six real numbers $A = \{a\_0, a\_1, \ldots a\_5\}$ with $a\_i < a\_{i+1}$. We construct a right-angled hexagon $H\_A$ in the complex plane by connecting $a\_0$ to $a\_1$, connecting $a\_2$ to $a\_3$, and connecting $a\_4$ to $a\_5$ by line segments, contained in the real axis. We then connect $a\_5$ to $a\_0$, connect $a\_1$ to $a\_2$, and connect $a\_3$ to $a\_4$ by arcs of circles orthogonal to the real axis, contained in the upper half plane. 


$H\_A$ appears to have six degrees of freedom, but these are reduced to just three after translating, scaling, and applying a Mobius transformation that preserves the real axis as well as $a\_0$ and $a\_5$. One must uniformize this hexagon (ie compute the hyperbolic lengths of the sides) to answer your question. I think this is a variant of the Schwarz-Christoffel transformation, but I failed to find a reference after looking for a bit. Perhaps look in Nehari's book "Conformal mappings".


That done, it is easy to double the hexagon across the real axis and apply a Mobius transformation to make the sides corresponding to $l\_2$ and $l\_3$ concentric. The log of the ratio of their radii now gives the desired modulus.


