Completeness of 𝑝-Weil-Petersson distance

Our goal of this paper is to research the completeness of the p p -Weil-Petersson distance, which is induced by the p p -Weil-Petersson metric on the p p -integrable Teichmüller space of hyperbolic Riemann surfaces. As a result, we see that the metric is incomplete for all the hyperbolic Riemann surfaces with Lehner’s condition except for the ones that are conformally equivalent to either the unit disk or the punctured unit disk. The proof is based on the one by Wolpert’s original paper, which is given in the case of compact Riemann surfaces.