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 ?
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 !