## 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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