Growth of the Weil-petersson Inradius of Moduli Space

In this paper we study the systole function along WeilPetersson geodesics. We show that the square root of the systole function is uniform Lipschitz on the Teichmüller space endowed with the Weil-Petersson metric. The inradius of a metric space is the largest radius of metric balls contained in the interior of the metric space. As applications of the result above, we study the growth of the Weil-Petersson inradius of the moduli space of Riemann surfaces of genus g with n punctures as a function of g and n. We show that the Weil-Petersson inradius is comparable to 1 in n, and is comparable to √ ln g in g. Moreover, we also study the asymptotic behavior, as g goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in the Teichmüller space. We show that they behavior like o(( 1 g )(3− ) as g → ∞, where is an arbitrary positive number.