Determinant of Laplacians on Heisenberg manifolds

✍ Scribed by Kenro Furutani; Serge de Gosson

• Book ID: 104343236
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 288 KB
• Volume: 48
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(03)00053-6

✦ Synopsis

We give an integral representation of the zeta-regularized determinant of Laplacians on threedimensional 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.