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MiS Preprint
59/2006

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

Yulia Klochko and Alexey Kokotov

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 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 this moduli space. 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.

Received:
Jul 3, 2006
Published:
Jul 3, 2006
Keywords:
polyhedral surfaces, determinants of Laplacians, quadratic differentials

Related publications

inJournal
2007 Repository Open Access
Yulia Klochko and Alexey Kokotov

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

In: Manuscripta mathematica, 122 (2007) 2, pp. 195-216