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On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric. (English) Zbl 0606.32014

Using the methods and original approach to Teichmüller theory developed by the author and A. Fischer [Math. Ann. 267, 311-345 (1984; Zbl 0518.32015); Trans. Am. Math. Soc. 284, 319-335 (1984; Zbl 0578.58006); J. Reine Angew. Math. 352, 151-160 (1984; Zbl 0573.32021)] the author obtains a formula for sectional curvature of Teichmüller space T(p) with respect to its Weil-Petersson metric in terms of the Laplace- Beltrami operator on functions (Scott Wolpert has derived independently a similar result). He shows that the sectional curvature as well as the holomorphic sectional curvature and Ricci curvature are negative [see L. V. Ahlfors, J. Anal. Math. 9, 161-176 (1961; Zbl 0148.312)]. Bounds on the holomorphic and the Ricci curvature are given; in particular, the holomorphic sectional curvature of T(p) (p is the genus of the surface, \(p>1)\) is strictly negative and bounded by \(-1/4\pi (p- 1).\)
Reviewer: B.N.Apanasov

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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References:

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[2] Ahlfors, L.V.: Curvature properties of Teichmüller’s space, J. Analyse Math. 9, 161-176 (1961) · Zbl 0148.31201
[3] Ahlfors, L.V.: The complex analytic structure on the space of closed Riemann surfaces. In Analytic functions, Princeton Univ. Press (1960) · Zbl 0100.28903
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[9] Fischer, A. E., Tromba, A. J.: On the Weil-Petersson metric on Teichmüller space, Trans. AMS Vol. 284, No. 1, July 1984 · Zbl 0578.58006
[10] Fischer, A. E., Tromba, A. J.: Almost complex principle fibre bundles and the complex structure on Teichmüller space, Crelles J., Band 352 (1984) · Zbl 0573.32021
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