Growth of the Weil-Petersson inradius of moduli space

Abstract

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of 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 $\sqrt{\ln g}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in Teichmüller space. We show that they behave like $o((\frac{1}{g}{)}^{(3-\epsilon )g})$ as $g\to \infty$, where $\epsilon >0$ is arbitrary.

Annales de l'Institut Fourier, to appear

References in corpus (1)

• Geometry of the Weil-Petersson completion of Teichmüller space

Cited by in corpus (5)

• Large genus asymptotics for lengths of separating closed geodesics on random surfaces
• Small eigenvalues of closed Riemann surfaces for large genus
• A new uniform lower bound on Weil-Petersson distance
• Maximal systole of hyperbolic surface with largest $S^3$ extendable abelian symmetry
• Shape of filling-systole subspace in surface moduli space and critical points of systole function