The Curvature of the Petersson-Weil Metric on the Moduli Space of Kähler-Einstein Manifolds
The Petersson-Weil metric is a main tool for investigating the geometry of moduli spaces. When A. Weil considered the classical Teichmüller space from the viewpoint of deformation theory, he suggested, in 1958, investigating the Petersson inner product on the space of holomorphic quadratic differentials. He conjectured that it induced a Kähler metric on the Teichmüller space. After proving this property, Ahlfors showed, in 1961, that the holomorphic sectional and Ricci curvatures were negative. Royden’s conjecture of a precise upper bound for the holomorphic sectional curvature was proven by Wolpert and Tromba in 1986 along with the negativity of the sectional curvature.
KeywordsVector Field Modulus Space Horizontal Lift Holomorphic Sectional Curvature Holomorphic Vector Field
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