Abstract
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 \({\mathcal Q_1(1, \dots, 1, [-1]^L)}\) 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 \({\mathcal Q_1(1, \dots, 1, [-1]^L)}\). 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.
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Klochko, Y., Kokotov, A. Genus one polyhedral surfaces, spaces of quadratic differentials on tori and determinants of Laplacians. manuscripta math. 122, 195–216 (2007). https://doi.org/10.1007/s00229-006-0063-1
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DOI: https://doi.org/10.1007/s00229-006-0063-1