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Holomorphic Factorization of Determinants of Laplacians using Quasi-Fuchsian Uniformization
Authors:Andrew Mcintyre  Lee-Peng Teo
Institution:(1) Bennington College, One College Drive, Bennington, Vermont 05201, USA;(2) Faculty of Information Technology, Multimedia University, Jalan Multimedia, Cyberjaya, 63100 Selangor Darul Ehsan, Malaysia
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 $$\phi_{1},\dotsc,\phi_{2d}$$ for ker Δ n . We prove that det$$\Delta_{n}/\det \langle\phi_{j},\phi_{k}\rangle$$ , 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.
Keywords:Holomorphic factorization  Laplacian  Period matrix  Differentials  Quasi-Fuchsian
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