Mayer-Vietoris formula for determinants of elliptic operators of Laplace-Beltrami type (after Burghelea, Friedlander and Kappeler)
✍ Scribed by Yoonweon Lee
- Book ID
- 104358297
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 970 KB
- Volume
- 7
- Category
- Article
- ISSN
- 0926-2245
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✦ Synopsis
The purpose of this note is to provide a short cut presentation of a Mayer-Vietoris formula due to Burghelea-Friedlander-Kappeler for the regularized determinant in the case of elliptic operators of Laplace
Reltrami type in the form typically needed in applications to torsion. Kevnx&.s: Regularized determinants, Dirichlet (Neumann', boundary condition. MS clu.ssificution: 58615. 58626. 1. Statement of Mayer-Vietoris formula for determinants Let (M. ,g) be a closed oriented Riemannian manifold of dimension d and r be an oriented submanifold of codimension I. We denote by u the unit normal vector field along T. Let Mr be the compact manifold with boundary P+ LI P-obtained by cutting M along r , where I? and r ~ arc copies of T and denote by p : Mr + M the identification map. The vector field u has the lift on MI7 which we denote by u again. Denote by rt the component of the boundary where the lift ot' v points outward. Given a smooth vector bundle E -+ M, denote by Et-the pull back of E + A4 to Mr by p. Let A : Cm(E) + Cm(E) be an elliptic, essentially self-adjoint, positive definite differential operator of Laplace-Beltrami type, where we say that A is of Laplace--Beltrami type if A is an operator of order 2 whose principal symbol is o~,(x. c) = IIt 11' Id.,. Id, E Et&(E,, E,). We denote by Ar : C"( El-) -+ P( Er) the extension of A to smooth sections of E,,. Consider Dirichlet and Neumann boundary conditions B, C on Tf LI I'-defined as follows: B : C"(E,-) -+ C"(Erlr-ur-), B(f) = flr+ur 7 C : Ca'(E,-) + Cm(Erlftur-), C(f) = ~(,f>lr-ur . Consider ;\r,H_= (Ar, B) : C"(Er) + P(E,-) @ Cm(ErJrtur-). From the properties of 4 it follows that Ar,b is invertible. Therefore we can define the corresponding Poisson operator