1. How exactly do we put hyperbolic structures on a sphere with cone point singularities. Should I consider that sphere with cone points as an extended complex plane with punctures endowed with a suitable metric of curvature -1 ?
2. I am having trouble with the following questions when I was reading a paper : the paper mentioned the facts I am looking for a proof without proof : Could somebody give me the proofs ?


a) Let us take a sphere S with 5 cone points ( you can assume that the cone points are zero such that we get a sphere with 5 cusps ) . Let F denote the family of all simple closed non-trivial geodesics $ \delta $ such that each connected component of $ S \backslash \delta $ contains all least 2 cone points. Now choose $ \gamma $ such that $\gamma $ minimizes the length among all $ \delta $ 's in F. Cut S open along $\gamma $ and call the two pieces $ S\_1 $ and $ S\_2 $ .Choose any two points p and q on $ \gamma $. Prove that : there exists another geodesic path c joining p and q in $S\_2$ such that c is not a subpath of $\gamma $.


b) Call $\gamma $ and $ \gamma' $ the two subpaths of $ \gamma $ separated by p and q.Prove that : ( or explain ) Either the concatenation of c with $\gamma $ or the concatenation of c with $\gamma' $ is is a simple closed curve whose simple closed geodesic representatives lie in F. [ my question is about "lying in F" part ].


c) Why do the above two facts imply that $ l(c) \geq $ minimum of l($\gamma$) ,l($\gamma'$) ?


I know these questions might be a bit technical, but I must apprecaiate your answer ,Thanks !



Questions about hyperbolic structures on a sphere with cone point singularities 
Concerning your first question, if you're used to Riemann surfaces then one way to understand hyperbolic metrics with cone singularities on the sphere is through a version of the uniformization theorem: given a closed Riemann surface $S$ with marked points $p\_1, \cdots, p\_n$ and angles $\theta\_1,\cdots,\theta\_n\in (0,2\pi)$ satisfying the obvious condition coming from the Gauss-Bonnet relation, there is a unique metric on $S$ compatible with the Riemann surface structure, with cone singularities of angle $\theta\_i$ at each $p\_i$.


This is a special case of Theorem A in:
Troyanov, Marc, 
*Prescribing curvature on compact surfaces with conical singularities.*
Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. 


In other terms, once you have chosen the $n$ angles in $(0,2\pi)$ with and $\sum\_i (2\pi-\theta\_i)>4\pi)$, hyperbolic metrics on the sphere with cone points of angles equal to the $\theta\_i$ are in one-to-one correspondence with configurations of $n$ distinct points on $CP^1$.


