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

Smoothness and strongly pseudoconvexity of p-Weil–Petersson metric

“…The proof is given similarly to Proposition 2.4 in [20]. For every subgroup Γ ′ of Γ, the Banach space L ∞ (H, Γ) is a subspace of L ∞ (H, Γ ′ ) since their L ∞ -norms coincide in L ∞ (H, Γ).…”

“…By estimation (3.8) in [20], µ(t)| N (t,r) p converges to 0 uniformly with respect to t as r → 0. Hence I 1 converges absolutely and uniformly with respect to t.…”

“…It follows that A forms an atlas of T p (Γ) for every p ≥ 2 and every Fuchsian group with Lehner's condition. We omit the proof since it is given almost similarly to the proof of Theorem 2.3 in [20] by replacing the letter "2" with "p" and by the post-composition of λ τ = β τ • R τ by χ.…”

“…In the recent paper [20], the author proved that the Kählerity and negativity of the Weil-Petersson metric on the square integrable Teichmüller space, which was proved originally for the Teichmüller space of a compact Riemann surface (cf. [2]).…”

Comparison Moduli Spaces of Riemann Surfaces

“…The convergent Weil-Petersson metric was obtained independently of Radnell, Schippers and Staubach[50,51]. Yanagishita[80] also showed that the complex structure from harmonic Beltrami differentials is compatible with the complex structure from the Bers embedding. When combined with the results of[51], this apparently shows that these two complex structures are equivalent to that obtained from fibrations over the compact surfaces for surfaces of type (g, n).…”