Curvature of the Weil-Petersson Metric in the Moduli Space of Compact Kähler-Einstein Manifolds of Negative First Chem Class

Abstract

For compact Riemann surfaces of genus at least two, using Petersson’s Hermitian pairing for automorphic forms, Weil introduced a Hermitian metric for the Teiclmuller space, now known as the Weil-Petersson metric. Ahlfors [1,2] showed that the Weil-Petersson metric is Kahler and that its Ricci and holomorphic section curvatures are negative. By using a different method of curvature computation, Royden [8] later showed that the holomorphic sectional curvature of the Weil-Petersson metric is bounded away from zero and conjectured the best bound to be \(- \frac{1}{{2\pi \left( {g - 1} \right)}}\) , where g is the genus. Recently Wolpert [12] and also Royden proved Royden’s conjecture on the bound of the holomorphic sectional curvature and obtained in addition the negativity of the Riemannian sectional curvature. Wolpert’s method used some SL(2,R) invariant first-order differential operators obtained by Maass [7]. Royden’s computation is based on the fact that the Pbincare metric on a compact Riemann surface of genus at least two is Einstein.