The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$.

The Teichmüller space $T_g$ also has a natural complex structure.

Could someone suggest a reference where the complex structure is given in terms of Fenchel-Nielsen coordinates?

A more rough question is: very naively I would combine $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$ into $\mathbb{H}^{3g-3}$ where $\mathbb{H}$ is the upper half plane. In the natural complex structure, is $T_g$ the same as $\mathbb{H}^{3g-3}$ as complex manifolds? I guess it's not, but is it proved somewhere?