Determinant of Laplacians on Heisenberg manifolds

• Published in: Journal of geometry and physics Vol. 48; no. 2; pp. 438 - 479
• Main Authors: Furutani, Kenro, de Gosson, Serge
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2003
• Subjects: Heat kernel; Heisenberg group; Kronecker’s second limit formula; Laplacian; Modified Bessel function; Poisson’s summation formula; Zeta-regularized determinant; Heisenberg group; Zeta-regularized determinant; Modified Bessel function; Poisson’s summation formula; Heat kernel; Kronecker’s second limit formula; Laplacian
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/S0393-0440(03)00053-6

We give an integral representation of the zeta-regularized determinant of Laplacians on three-dimensional Heisenberg manifolds, and study a behavior of the values when we deform the uniform discrete subgroups. Heisenberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typical fiber, then the deformation of the uniform discrete subgroups means that the “radius” of the fiber goes to zero. We explain the lines of the calculations precisely for three-dimensional cases and state the corresponding results for five-dimensional Heisenberg manifolds. We see that the values themselves are of the product form with a factor which is that of the flat torus. So in the last half of this paper we derive general formulas of the zeta-regularized determinant for product type manifolds of two Riemannian manifolds, discuss the formulas for flat tori and explain a relation of the formula for the two-dimensional flat torus and the Kronecker’s second limit formula.