The Weil-Petersson geometry of the moduli space of Riemann surfaces

Teo, Lee-Peng

doi:10.1090/S0002-9939-08-09692-5

The Weil-Petersson geometry of the moduli space of Riemann surfaces
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Proc. Amer. Math. Soc. 137 (2009), 541-552 Request permission

Abstract:

In 2007, 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 the 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 Teichmüller space equipped with a Hilbert structure induced by the Weil–Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.

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