The Teichmüller space Teich ( S ) of a surface S in genus g > 1 is a totally real submanifold of the quasifuchsian space QF ( S ). We show that the determinant of the Laplacian det ' (Δ) on Teich ( S ) has a unique holomorphic extension to QF ( S ). To realize this holomorphic extension as the determinant of differential operators on S , we introduce a holomorphic family {Δμ,ν} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S . We study the determinant of this family {Δμ,ν} and show how this family realizes the holomorphic extension of det ' (Δ) as its determinant.
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