DETERMINANTS OF LAPLACIANS ON THE SPACE OF CONICAL METRICS ON THE SPHERE

On a compact surface with smooth boundary, the determinant of the Lapl acian associated to a smooth metric on the surface (with Dirichlet bou ndary conditions if the boundary is nonempty) is a well-defined isospe ctral invariant. As a function on the moduli space of such surfaces, i t is a smooth function whose boundary behavior in certain cases is wel l understood; see [OPS and K]. In this paper, we restrict ourselves to a certain class of singular metrics on closed surfaces called conical metrics. We show that the determinant of the associated Laplacian is still well defined and that it is a real analytic function on a suitab ly restricted subset of the space of conical metrics on the sphere.