Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization

Abstract

For a quasi-Fuchsian group $\Ga$ with ordinary set $\Omega$, and $\Delta_{n}$ the Laplacian on \n differentials on $\Ga\bk\Omega$, we define a notion of a Bers dual basis $\phi_{1},...c,\phi_{2d}$ 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 \cite{TT1}, the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta 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 \cite{MT}.