1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex curves ?
2) Is there such a way to characterise the convergence of a sequence of translation surface towards a degenerate translation surface(which is an abelian differential over a stable curve) in term of flat metrics on the underlying topological surface, still for the Deligne Mumford compactification ?
