Genus one polyhedral surfaces,spaces of quadratic differentials on tori and determinants of Laplacians

We prove a formula for the determinant of the Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray–Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also considered. We study the determinant of the Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeros and derive formulas for variations of this functional with respect to natural coordinates on . We give also a new proof of Troyanov’s theorem stating the existence of a conformal flat conical metric on a compact Riemann surface of arbitrary genus with a prescribed divisor of conical points.