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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


Author: Yoonweon Lee
Journal: Trans. Amer. Math. Soc. 355 (2003), 4093-4110
MSC (2000): Primary 58J52, 58J50
DOI: https://doi.org/10.1090/S0002-9947-03-03249-5
Published electronically: June 24, 2003
MathSciNet review: 1990576
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Abstract | References | Similar Articles | Additional Information

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.


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  • [1] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. MR 53:1655a
  • [2] B. Booß-Bavnbek, and K.P. Wojciechowski, Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, 1993. MR 94h:58168
  • [3] D. Burghelea, L. Friedlander and T. Kappeler, Mayer-Vietoris type formula for determinants of elliptic differential operators, J. of Funct. Anal. 107 (1992), 34-66. MR 93f:58242
  • [4] D. Burghelea, L. Friedlander , T. Kappeler and P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules, Geom. and Funct. Anal. 6 (1996), 751-859. MR 97i:58177
  • [5] S. Cappell, R. Lee and E. Miller, Self-adjoint elliptic operators and manifold decompositions, Part I: Low eigenmodes and stretching, Comm. Pure Appl. Math. 96 (1996), 825-866. MR 97g:58162
  • [6] S. Klimek, K.P. Wojciechowski, Adiabatic cobordism theorems for analytic torsion and $\eta $-invariant, J. of Funct. Anal. 136 (1996), 269-293. MR 97a:58195
  • [7] Y. Lee, Mayer-Vietoris formula for the determinant of a Laplace operator on an even dimensional manifold, Proc. Amer. Math. Soc. 121-6 (1995), 1933-1940. MR 95g:58255
  • [8] Y. Lee, Mayer-Vietoris formula for the determinants of elliptic operators of Laplace-Beltrami type (after Burghelea, Friedlander and Kappeler), Diff. Geom. and Its Appl. 7 (1997), 325-340. MR 99a:58163
  • [9] 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.
  • [10] 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.
  • [11] D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-209. MR 45:4447
  • [12] A. Voros, Spectral function, special functions and Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465. MR 89b:58173

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Additional Information

Yoonweon Lee
Affiliation: Department of Mathematics, Inha University, Inchon, 402-751, Korea
Email: ywlee@math.inha.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-03-03249-5
Keywords: Zeta-determinant, gluing formula, Laplacian, Dirichlet (Neumann) boundary condition, absolute (relative) boundary condition, adiabatic decomposition
Received by editor(s): April 15, 2002
Received by editor(s) in revised form: October 10, 2002
Published electronically: June 24, 2003
Additional Notes: The author was partially supported by Korea Research Foundation Grant KRF-2000-015-DP0045
Article copyright: © Copyright 2003 American Mathematical Society

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