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The Burghelea-Friedlander-Kappeler-gluing formula for zeta-determinants on a warped product manifold and a product manifold. (English) Zbl 1334.58022

J. Math. Phys. 56, No. 12, 123501, 19 p. (2015); erratum ibid. 59, No. 1, 019902, 1 p. (2018).
Let \(M\) be a closed Riemannian manifold and \(\Delta_M\) the Laplacian on smooth functions on \(M\). The determinant \(\det(\Delta_M)\) is defined via zeta-regularisation: informally this means that \(\log\det(\Delta_M) = \zeta'(0)\) where \[ \zeta(s) = \sum_{j > 0} \lambda_j^{-s} \] is the spectral zeta-function of \(M\) (the series above converges in a half-plane and meromorphically extends to \(\mathbb C\)). A similar definition holds when \(M\) has boundary and one considers Dirichlet, Neumann or mixed conditions there. The Burghelea-Kappeler-Friedlander formula computes the difference \[ \log\det(\Delta_M) - \log\det(\Delta_{M_1}) - \log\det(\Delta_{M_2}) \] when \(M = M_1 \cup_N M_2\), \(N\) being a smooth separating hypersurface in \(M\). This can be useful to compute the determinant of \(\Delta_M\) from those of the smaller pieces \(\Delta_{M_1}\) and \(\Delta_{M_2}\), or on manifolds with totally geodesic boundary by applying the formula to the double.
The formula is not completely explicit in general: there are two terms depending on the spectrum of \(N\) and the restriction of harmonic functions on \(M\) to \(N\), and a third that has no closed expression in general. The latter has been computed when \(M\) has a product structure near \(N\). In this paper the authors give an explicit formula in the more general case where \(M\) has only a warped product formula near \(N\). The new ingredient in the formula is the determinant on warped product cylinders \(N\times I\) where \(I\) is a closed interval.

MSC:

58J52 Determinants and determinant bundles, analytic torsion
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35C20 Asymptotic expansions of solutions to PDEs
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
58J05 Elliptic equations on manifolds, general theory
58J26 Elliptic genera

Citations:

Zbl 1272.58016
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Full Text: DOI

References:

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