Holomorphic extensions of Laplacians and their determinants

The Teichmuller space Teich(S) of a surface S in genus g > I is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(A) 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 (Delta(mu,v)) 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 (Delta(mu,v)) and show how this family realizes the holomorphic extension of det'(A) as its determinant. (C) 2006 Elsevier Inc. All rights reserved.