Smoothness and strongly pseudoconvexity of p-Weil–Petersson metric

The Teichmüller space of a Riemann surface of analytically finite type has a complex structure modeled on the complex Hilbert space consisting of harmonic Beltrami differentials on the surface equipped with hyperbolic L-norm. The Weil– Petersson metric is an Hermitian metric induced by this Hilbert manifold structure and is studied in many fields. In the complex analysis, Ahlfors [2, 3] proved that the Weil–Petersson metric is a Kähler metric and has the negative holomorphic sectional curvature, negative Ricci curvature and negative scalar curvature. In the hyperbolic geometry, Wolpert [17, 18] gave the several relations between the Weil–Petersson metric and the Fenchel–Nielsen coordinate. In general, that Hilbert manifold structure cannot be introduced to the Teichmüller space of a Riemann surface of analytically infinite type (cf. [9]). Takhtajan and Teo [15] realized this structure as a distribution on the universal Teichmüller space. Cui [5] accomplished the same result on the subset of the universal Teichmüller space independently of Takhtajan and Teo. Hui [6] and Tang [16] extended the argument of Cui to the subset modeled on p-integrable Beltrami differentials for p ≥ 2, which we call the p-integrable Teichmüller space. Later, Radnell, Schippers and Staubach [11, 12, 13] composed a Hilbert manifold structure on a certain refined Teichmüller space of a bordered Riemann surface, which is refered to as the WP-class Teichmüller space. In [5, 15], the Weil–Petersson metric was studied for each Hilbert manifold structure. In particular, it was shown that this metric is negatively curved (cf. [15]) and complete (cf. [5]). Recently, Matsuzaki [8] researched some properties of the p-Weil– Petersson metric on the p-integrable Teichmüller space of the unit disk for p ≥ 2. This metric is a certain extended concept of the Weil–Petersson metric on the square integrable Teichmüller space. In fact, it was proved in [8] that the metric is complete and continuous. Based on their results, the author [19] introduced some complex analytic structure on the p-integrable Teichmüller space of a Riemann surface with Lehner’s condition