Holomorphic Extensions of Laplacians and Their Determinants

The Teichmueller 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’(\Delta) 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,\nu}} 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,\nu}} and show how this family realizes the holomorphic extension of det’(\Delta) as its determinant.