Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization

Abstract

For a quasi-Fuchsian group Γ with ordinary set Ω, and Δ_n the Laplacian on n-differentials on Γ\Ω, we define a notion of a Bers dual basis ϕ_1,…,ϕ_2_d for ker Δ_n. We prove that det Δ_n/det⟨ϕ_j,ϕ_k⟩, is, up to an anomaly computed by Takhtajan and the second author in (Commun. Math Phys 239(1-2):183–240, 2003), the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta function Z(n). This generalizes the D’Hoker–Phong formula det Δ_n=c_g,_nZ(n), and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in Analysis 16, 1291–1323, 2006.