Determinant of Laplacians on Heisenberg manifolds

[摘要] 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. (C) 2003 Elsevier Science B.V. All rights reserved.

[发布日期] 2003-11-01

[关键词] Heisenberg group;zeta-regularized determinant;Laplacian;heat kernel;Kronecker's second limit formula;modified bessel function;Poisson's summation formula