A new uniform lower bound on Weil–Petersson distance

In this paper we study the injectivity radius based at a fixed point along Weil–Petersson geodesics. We show that square root of is 0.3884-Lipschitz on Teichmüller space endowed with metric. As an application reprove systole function uniformly Lipschitz metric, where constant can be chosen to 0.5492. Applications asymptotic geometry moduli Riemann surfaces for large genus will also discussed.