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 ?