Determinant of Laplacians on Heisenberg manifolds

Abstract

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 Kroneker's second limit formula.

Determinants of pseudo-Laplacians

“…We will make use the well-known identity (see, e.g., [2] is the theta-function. and X = T 3 are given in [13] and [6].…”

“…The zeta-regularized determinant of Laplacian on a compact Riemannian manifold was introduced in [11] and since then was studied and used in an immense number of papers in string theory and geometric analysis, for our future purposes we mention here the memoir [5], where the determinant of Laplacian is studied as a functional on the space of smooth Riemannian metrics on a compact two-dimensional manifold, and the papers [6,13], where the reader may find explicit calculation of the determinant of Laplacian for three-dimensional flat tori and for the sphere S 3 (respectively). The main result of the present paper is a comparison formula relating det(Δ α,P − λ) to det(Δ − λ), for λ ∈ C \ (Spectrum(Δ) ∪ Spectrum(Δ α,P )) (see Theorem 1 in Section 4 and Theorem 2 in Section 5).…”

Zeta determinants on cones and products

“…For in the case of a product manifold one can write the regular term in the Mellin transform of the heat function by adding and subtracting the singular part of the integrand (see for example [15,Section 3]), since this singular part is known (it corresponds to the product of the expansions of the singular parts of the heat kernels of the factors). This approach provides a formula for the regularized determinant involving, in the regular part, a finite integral of some complicate function, and was used in a somehow formal way in [13]. Since some derivative of the logarithmic Gamma function Γ(−λ, S) is the Mellin Laplace transform of the heat function (see the proof of Proposition 2.7 of [20] for details), the result for ζ (0, S (0) ) given in Theorem 2.3 is an evaluation of the finite integrals appearing in the formulas given in Section 3 of [13].…”

“…This approach provides a formula for the regularized determinant involving, in the regular part, a finite integral of some complicate function, and was used in a somehow formal way in [13]. Since some derivative of the logarithmic Gamma function Γ(−λ, S) is the Mellin Laplace transform of the heat function (see the proof of Proposition 2.7 of [20] for details), the result for ζ (0, S (0) ) given in Theorem 2.3 is an evaluation of the finite integrals appearing in the formulas given in Section 3 of [13].…”

“…The case of a bounded cone was deeply investigated [1][2][3]7,10,11] because the analytic geometric setting is well known by seminal works of Cheeger [5,6]. The case of a product space is less studied, but see [13]. In all these works, the approach is based on the so called heat kernel methods (see for example [14]).…”