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

2006/05/23 by Yulia Klochko, Klochko, Yulia, Alexey Kokotov +1
Mathematics · #Advanced Algebra and Geometry #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0605614

openalex publication_date 2006/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 w. r. t. natural coordinates on this moduli space. A new proof of the Troyanov theorem about flat conical metrics on compact Riemann surfaces of arbitrary genus is also given.

Citations

• Further results on Functional Determinants of Laplacians in Simplicial Complexes

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