Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

Lee, Yoonweon

doi:10.1090/S0002-9947-03-03249-5

Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion
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by Yoonweon Lee

Trans. Amer. Math. Soc. 355 (2003), 4093-4110

DOI: https://doi.org/10.1090/S0002-9947-03-03249-5

Published electronically: June 24, 2003

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Abstract:

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

References

M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
Bernhelm Booß-Bavnbek and Krzysztof P. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233386, DOI 10.1007/978-1-4612-0337-7
D. Burghelea, L. Friedlander, and T. Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), no. 1, 34–65. MR 1165865, DOI 10.1016/0022-1236(92)90099-5
D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. Funct. Anal. 6 (1996), no. 5, 751–859. MR 1415762, DOI 10.1007/BF02246786
Sylvain E. Cappell, Ronnie Lee, and Edward Y. Miller, Self-adjoint elliptic operators and manifold decompositions. I. Low eigenmodes and stretching, Comm. Pure Appl. Math. 49 (1996), no. 8, 825–866. MR 1391757, DOI 10.1002/(SICI)1097-0312(199608)49:8<825::AID-CPA3>3.3.CO;2-4
Slawomir Klimek and Krzysztof P. Wojciechowski, Adiabatic cobordism theorems for analytic torsion and $\eta$-invariant, J. Funct. Anal. 136 (1996), no. 2, 269–293. MR 1380656, DOI 10.1006/jfan.1996.0031
Yoonweon Lee, Mayer-Vietoris formula for the determinant of a Laplace operator on an even-dimensional manifold, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1933–1940. MR 1254845, DOI 10.1090/S0002-9939-1995-1254845-7
Yoonweon Lee, Mayer-Vietoris formula for determinants of elliptic operators of Laplace-Beltrami type (after Burghelea, Friedlander and Kappeler), Differential Geom. Appl. 7 (1997), no. 4, 325–340. MR 1485640, DOI 10.1016/S0926-2245(96)00053-8
Y. Lee, Burghelea-Friedlander-Kappeler’s gluing formula and the adiabatic decomposition of the zeta-determinant of a Dirac Laplacian, to appear in Manuscripta Math.
P. Park and K. Wojciechowski with Appendix by Y. Lee, Adiabatic decomposition of the $\zeta$-determinant of the Dirac Laplacian I. The case of an invertible tangential operator, Comm. in PDE. 27 (2002), 1407-1435.
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), no. 3, 439–465. MR 891947