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

Abstract

Abstract. We prove a formula for the determinant of 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 Laplacian as a functional on the moduli space Q 1 (1, . . . , 1, [−1] L ) 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 Q 1 (1, . . . , 1, [−1] L ). 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.

Polyhedral surfaces and determinant of Laplacian

“…Here we generalize the results of [10], [8], [2] to the case of polyhedral surfaces of an arbitrary genus. The main result of the paper, the explicit formula for the determinant, is given by equation (30) below.…”

“…The flat conical metrics |ω| 2 considered in [10] are very special: the divisor of the conical points of this metric is not arbitrary (it belongs to the canonical class of divisors) and the conical angles at the conical points are integer multiples of 2π. Later, in [8], this restrictive condition has been eliminated in the case of polyhedral surfaces of genus one (it should be noted that the case of genus zero was studied in [2]).…”