Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :


By $ \bar{P} $ , we denote a topological pair of pants ( that is, a 2-sphere with three open disks removed ) with boundary, and by $P$, we would mean the same without any boundary. Let $\mathcal {Mod(\bar{P})}$ denote the space of all complete hyperbolic metric on $ \bar{P} $ which make the boundary components of $P$ geodesics ; call these geodesics $\gamma\_1, \gamma\_2, \gamma\_3 $ . Since these metrics are completely determined by the hyperbolic lengths $ l\_1, l\_2, l\_3 $ of the three boundary components, we can give $ \mathcal { Mod(\bar{P} ) } $ the co-ordinates $ l\_1, l\_2, l\_3 $ and so think of $ \mathcal { Mod(\bar{P} ) } $ as $\mathbb{R\_+}^3$. Consider the following map $ F : \mathcal {Mod(\bar{P})} \cong \mathbb{R\_+}^3 \to \mathbb{R\_+}^3 $ defined as follows : 


$m\_i$ is the module of the maximal ring domain in $ \bar{P} $ whose core curve is homotopic to $ \gamma\_i $, i.e. for all ring domain $R\_i$'s whose core curve is homotopic to $ \gamma\_i $, we have $ m\_i = sup mod (R\_i) $ and this supremum is achieved by a ring domain, which is the the pair of pants $\bar{P}$ minus the unique geodesic joining $ \gamma\_{i+1 modulo 3}, \gamma\_ {i+2 modulo 3} . $. Now define the map : $ F : \mathcal {Mod(\bar{P})} \to \mathbb{R\_+}^3 $ by $ F(l\_1,l\_2,l\_3) = (m\_1,m\_2,m\_3) $. I was wondering whether this map is a homeomoprhism of $ \mathbb{R\_+}^3 $. I guess constructing $F$ explicitly would be really hard. Is there any literature along this line ?



Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains
I think the map $F$ is injective, but not surjective. 


There is a unique conformal map of the interior of the pants to the complement of 3 slits in $\mathbb{RP}^1\subset \mathbb{CP}^1$, up to the action of $PGL\_2(\mathbb{R})$. If we parameterize the slits as $[a\_0,a\_1],[a\_2,a\_3],[a\_4,a\_5]$, where $a\_0 < a\_1 < a\_2 < a\_3 < a\_4 < a\_5$ in $\mathbb{R}$,
then the configuration is invariant under complex conjugation, which induces a hyperbolic isometry of the pants. So the pant seams are the intervals $[a\_1,a\_2], [a\_3,a\_4], [a\_5,a\_0]$ taken in cyclic order on the circle $\mathbb{RP}^1$. In fact, one may think of the map as "sewing" the cuffs of the pants along the two intervals connecting the seams. The rings $R\_i$ are the complements of the slits $[a\_i,a\_{i+1}]\cup [a\_{i+3},a\_{i+4}]$, indices taken $(\mod 6)$ (where $R\_i \cong R\_{i+3}$). The points are uniquely determined by the modulus of $R\_i$, up to the action of $PSL(2,\mathbb{R})$, or by the [cross-ratio](http://en.wikipedia.org/wiki/Cross-ratio) $[a\_i,a\_{i+1};a\_{i+3},a\_{i+4}]$. 


Therefore the modulus of each ring determines uniquely another ring $U\_i$ 
which is the complement of the slits $[a\_{i+1},a\_{i+3}]\cup [a\_{i+4},a\_i]$ (where the intervals are determined by the cyclic order and orientation of $\mathbb{RP}^1$). 
Then $R\_{i+1} \supset U\_i$, $R\_{i+2}\supset U\_i$, as homotopy equivalences. So the moduli of the rings $R\_{i+1}$ and $R\_{i+2}$ bound the modulus of the ring $U\_i$, by the monotonicity of moduli of annuli, which uniquely determines the modulus of $R\_i$. So this shows one cannot achieve all possible triples of moduli, so the map $F$ is not surjective. 


