The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different uniformizations in the math literature. I have several questions related to different uniformizations. I appreciate any comment or pointing to references.

**For a closed Riemann surface $\Sigma_{g,n}$ of genus $g$ with $n$ number of punctures:**

*Is there a Fuchsian uniformization in terms of a Fuchsian group $\Gamma$ acting on the upper half-plane ${\mathbb{H}}$?*I suppose there is a unique uniformization up to the action of $\text{PSL}(2,\mathbb{R})$, the group of orientation-preserving automorphisms of $\mathbb{H}$.

*Is there a Schottky uniformization in terms of a Schottky group $S$?*I suppose there is but I am not sure about the uniqueness. Any comment would help.

*Is there a way to get Schottky uniformization from a Fuchsian one or vice versa?**Is there a way to get objects like period matrix, prime form, abelian differential, and ... in terms of Fenchel-Nielsen coordinates?**Is there a way to get complex analytic structure of Teichmuller in terms of Fenchel-Nielsen coordinates?*

And the same set of questions for a bordered surfaces,

**For a bordered Riemann surface $\Sigma_{g,n}^{(L_1,\cdots,L_n)}$ of genus $g$ with $n$ number of boundaries of lenghts $\{L_1,\cdots,L_n\}$:**

*Is there a Fuchsian uniformization in terms of a Fuchsian group $\Gamma$ acting on the upper half-plane ${\mathbb{H}}$?*I suppose there is a unique uniformization up to the action of $\text{PSL}(2,\mathbb{R})$, the group of orientation-preserving automorphisms of $\mathbb{H}$.

*Is there a Schottky uniformization in terms of a Schottky group $S$?**Is there a way to get Schottky uniformization from a Fuchsian one or vice versa?**Is there a way to get objects like period matrix, prime form, abelian differential, and ... in terms of Fenchel-Nielsen coordinates?**Is there a way to get complex analytic structure of Teichmuller in terms of Fenchel-Nielsen coordinates?*I am not even sure that in this case, the Teichmuller space has a complex analytic structure.