The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3856–3891 | Cite as

The Polynomial Associated with the BFK-Gluing Formula of the Zeta-Determinant on a Compact Warped Product Manifold

  • Klaus KirstenEmail author
  • Yoonweon Lee


In the proof of the BFK-gluing formula of the zeta-determinant of a Laplacian there appears a polynomial of degree less than half of the dimension of an underlying manifold. This polynomial is determined completely by some data on a collar neighborhood of a cutting compact hypersurface. In this paper we compute the polynomial in terms of a warping function when a collar neighborhood of a cutting hypersurface is a warped product manifold. We also use a similar method to compute the values of a relative zeta function and a zeta function associated to the Dirichlet-to-Neumann operator at zero on a warped product manifold.


BFK-gluing formula Relative zeta-determinant Dirichlet-to-Neumann operator Warped product metric Warping function 

Mathematics Subject Classification

Primary: 58J20 Secondary: 14F40 


  1. 1.
    Bär, C.: Zero sets of solutions to semilinear elliptic systems of first order. Invent. Math. 138, 183–202 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bunke, U.: Relative index theory. J. Funct. Anal. 105, 63–76 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burghelea, D., Friedlander, L., Kappeler, T.: Mayer–Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107, 34–66 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carron, G.: Déterminant relatif et fonction Xi. Am. J. Math. 124, 307–352 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chernoff, P.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Forman, R.: Functional determinants and geometry. Invent. Math. 88, 447–493 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fucci, G., Kirsten, K.: The spectral zeta function for Laplace operators on warped product manifolds of type \(I \times _{f} N\). Commun. Math. Phys. 317, 635–665 (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd edn. CRC Press Inc, Boca Raton (1994)Google Scholar
  9. 9.
    Gilkey, P.B.: Asymptotic Formulae in Spectral Geometry. Chapman and Hall/CRC, Boca Raton (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, New York (1965)zbMATHGoogle Scholar
  11. 11.
    Kazdan, J.: Unique continuation in geometry. Commun. Pure Appl. Math. 41, 667–681 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kirsten, K.: Spectral Functions in Mathematics and Physics. Chapman and Hall/CRC, Boca Raton (2002)zbMATHGoogle Scholar
  13. 13.
    Kirsten, K., Lee, Y.: The Burghelea–Friedlander–Kappeler-gluing formula for zeta-determinants on a warped product manifold and a product manifold. J. Math. Phys. 58(12), 123501-1-19 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kirsten, K., Lee, Y.: The BFK-gluing formula and relative determinants on manifolds with cusps. J. Geom. Phys. 117, 197–213 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, Y.: Mayer–Vietoris formula for the determinant of a Laplace operator on an even-dimensional manifold. Proc. Am. Math. Soc 123(6), 1933–1940 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Müller, W.: Relative zeta functions, relative determinants, and scattering theory. Commun. Math. Phys. 192, 309–347 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Müller, J., Müller, W.: Regularized determinants of Laplace type operators, analytic surgery and relative determinants. Duke Math. J. 133, 259–312 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for Special Functions of Mathematical Physics. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  19. 19.
    Park, J., Wojciechowski, K.: Agranovich–Dynin formula for the zeta-determinants of the Neumann and Dirichlet problems. Contemp. Math. 366, 109–121 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Voros, A.: Spectral functions, special functions and Selberg zeta function. Commun. Math. Phys. 110, 439–465 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsBaylor UniversityWacoUSA
  2. 2.Department of MathematicsInha UniversityIncheonKorea

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