The analytic torsion of the finite metric cone over a compact manifold

TL;DR

This paper derives an explicit formula for the $L^2$ analytic torsion of finite metric cones over compact manifolds, interpreting its components and analyzing its behavior under boundary collapsing.

Contribution

It provides a new explicit formula for the analytic torsion of finite metric cones and interprets the factors involved, including the regularizing component.

Findings

Explicit formula for $L^2$ analytic torsion of finite metric cones

Interpretation of factors in the torsion formula

Analysis of boundary collapsing limit behavior

Abstract

We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.