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The Riemannian sectional curvature operator of the Weil-Petersson metric and its application. (English) Zbl 1295.30103

Let \(S\) be a closed surface of genus \(g\) where \(g > 1\), and \(T_g\) be the Teichmüller space of \(S\). \(T_g\) carries various metrics which have respective properties. For example the Teichmüller metric is a complete Finsler metric. The McMullen metric, Ricci metric and perturbed Ricci metric have bounded geometry. The Weil-Petersson metric is Kähler and incomplete. There are also some other metrics on \(T_g\) like the Bergman metric, Carathéodory metric, Kähler-Einstein metric, Kobayashi metric and so on. In this paper the author focusses on the Weil-Petersson case. Let us denote by \(\mathrm{Teich}(S)\) the space \(T_g\) endowed with the Weil-Petersson metric. The geometry of the Weil-Petersson metric has been well studied in the past decades. The curvature aspect of \(\mathrm{Teich}(S)\) is also very interesting and plays an important role in the geometry of the Weil-Petersson metric. In this paper the author shows that the Riemannian sectional curvature operator of \(\mathrm{Teich}(S)\) is non-positive definite. Also, as an application, the author shows that any twist harmonic map from the rank-one hyperbolic spaces \(H_{Q,m} =Sp(m,1)/Sp(m)\cdot Sp(1)\) or \(H_{O,2} = F^{20}_4/SO(9)\) into \(\mathrm{Teich}(S)\) is a constant.

MSC:

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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