A decomposition for plane domains with the quasihyperbolic metric

It is known that complete Riemannian surfaces can be obtained by pasting three kinds of pieces. In this paper we prove an analogous result in the context of plane domains with their quasihyperbolic metrics. In order to do it, we prove several facts about quasihyperbolic closed geodesics of independent interest; for instance, we characterize the existence of quasihyperbolic minimizers, and we show that images of local quasihyperbolic geodesics are finite graphs.