Analytic surgery of the zeta function

TL;DR

This paper investigates how the zeta function of Laplace-type operators behaves asymptotically under geometric deformations called analytic surgery, especially when manifolds are stretched to form infinite cylindrical ends.

Contribution

It extends the theory of analytic surgery to the asymptotic analysis of zeta functions of Laplace-type operators on stretched manifolds.

Findings

Derived asymptotic formulas for zeta functions under manifold stretching.

Connected the behavior of zeta functions to geometric deformations.

Extended analytic surgery techniques to new spectral invariants.

Abstract

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of "analytic surgery", developed originally by Hassell, Mazzeo and Melrose \cite{mazz95-5-14,hass95-3-115,hass98-6-255} in the context of eta invariants and determinants.