The Weil--Petersson geometry of the moduli space of Riemann surfaces
In [4], Z. Huang showed that in the thick part of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the sectional curvature of the Weil--Petersson metric is bounded below by a constant depending on injectivity radius, but independent of the genus $g$. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichm\"uller space equipped with Hilbert structure induced by Weil--Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.