Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If I add a puncture somewhere, the new surface again has a unique hyperbolic metric of curvature $-1$. Is there a way to see how this metric changes from the previous one, e.g. in terms of Fenchel–Nielsen coordinates?
Since it is a very hard questions, I add a more concrete situation I am interested in.
Question 2: Suppose that $S$ instead comes equipped with a family of hyperbolic metrics, and a pants decomposition, so that one geodesic length tends to zero (so there is a long collar cylinder nearby). Suppose also that all the other Fenchel Nielsen coordinates are fixed. Let me add a puncture on the shrinking geodesic. I would like to understand why this does not cause other Fenchel-Nielsen coordinates to change much (and that the only change is that a new pair-of-pants appears instead of this vanishing geodesic, but all the rest will be intact in the limit).