Kählerity and Negativity of Weil–Petersson Metric on Square Integrable Teichmüller Space

The Weil–Petersson metric is a Hermitian metric originally defined on finite-dimensional Teichmüller spaces. Ahlfors proved that this metric is a Kähler metric and has some negative curvatures. Takhtajan and Teo showed that this result is also valid for the universal Teichmüller space equipped with a complex Hilbert manifold structure. In this paper, we stated that the Weil–Petersson metric can be also defined on a Hilbert manifold contained in the Teichmüller space of Fuchsian groups with Lehner’s condition, which we call the square integrable Teichmüller space, and proved that the results given by Ahlfors, Takhtajan, and Teo also hold in that case. Many parts of the proof were based on their ones. However, we needed more careful estimations in the infinite-dimensional case, which was achieved by two complex analytic characterizations of Lehner’s condition, by a certain integral equality for the partition of the upper half-plane by a Fuchisian group and by the invariant formula for the Bergman kernel.