Holomorphic Factorization of Determinants of Laplacians using Quasi-Fuchsian Uniformization - Mcintyre, Andrew; Teo, Lee-Peng (SPRINGER, 2008)
For a quasi-Fuchsian group Gamma with ordinary set Omega, and Delta(n) the Laplacian on n-differentials on GammaΩ, we define a notion of a Bers dual basis phi(1),...,phi(2d) for ker Delta(n). We prove that det Delta(n)/det <phi(j), phi(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 Delta(n) = c(g,n) Z(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.