Growth of the Weil-Petersson inradius of moduli space (Q2421942)

For a surface \(S_{g,n}\) of genus \(g\) with \(n\) punctures satisfying \(3g-3+n>0\), let Teich(\(S_{g,n}\)) be the Teichmüller space of \(S_{g,n}\) with the Weil-Petersson metric and let \(\mathcal{M}_{g,n}\) be the moduli space of \(S_{g,n}\) defined as the quotient of Teich\((S_{g,n})\) by the mapping class group Mod(\(S_{g,n}\)). The author studies the asymptotic behavior of the inradius of \(\mathcal{M}_{g,n}\) either as \(g\to \infty\) or \(n\to\infty\). It is proved that for all \(n\ge 0\) and \(g\ge 2\), the inradius of \(\mathcal{M}_{g,n}\) is comparable to \(\sqrt{\ln g}\) by a constant independent of \(g\); for all \(g\geq 0\) and \(n\geq 4\), it is comparable to 1 by a constant independent of \(n\). Here, the inradius is defined by the maximum of the Weil-Petersson distances dist\({}_{wp}(X, \partial\overline{\mathcal{M}}_{g,n})\) among all \(X\in \mathcal{M}_{g,n}\). To prove these results, the author considers the systole function \(\ell_{sys}\) on Teich(\(S_{g,n}\)) and gives a key theorem which establishes Lipschitz continuity of the square root of \(\ell_{sys}\) with respect to the Weil-Petersson distance, where the Lipschitz constant can be taken independently of \(g\) and \(n\). For the proof of Lipschitz continuity the author uses a thin-thick decomposition of the Weil-Petersson geodesics connecting two points in Teich(\(S_{g,n}\)) and estimates the norm of the gradient of the square root of geodesic length functions. \par Let \(\mathcal{M}_{g,n}^{\geqslant \epsilon}\) denote the \(\epsilon\)-thick part of of \(\mathcal{M}_{g,n}\). The moduli space \(\mathcal{M}_{g,n}\) is foliated by \(\partial\mathcal{M}_{g,n}^{\geqslant \epsilon}\) for all \(\epsilon\). The author shows that for any \(s>t\geq 0\) the Weil-Petersson distance between \(\partial\mathcal{M}_{g,n}^{\geqslant s}\) and \(\partial\mathcal{M}_{g,n}^{\geqslant t}\) is comparable to \(\sqrt{s}-\sqrt{t}\) by a constant independent of \(g\) and \(n\). \par Another interesting result in this paper is that for a closed surface \(S_g\) the author shows the asymptotic behavior of the Weil-Petersson volume of geodesic balls as \(g\to \infty\), where the geodesic balls have a finite radius and are away from the boundary of the completion of Teich(\(S_g\)).