Resolvent, heat kernel and torsion under degeneration to fibered cusps

TL;DR

This paper investigates the spectral behavior of the Hodge Laplacian on manifolds degenerating to fibered cusps, providing detailed asymptotics and linking analytic torsion to topological invariants.

Contribution

It offers new precise asymptotic formulas for the resolvent, heat kernel, and Laplacian determinant during degeneration, and relates analytic torsion to R-torsion in this setting.

Findings

Asymptotic formulas for resolvent, heat kernel, and Laplacian determinant

Topological description of analytic torsion in fibered cusp degeneration

Connection between analytic torsion and R-torsion

Abstract

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.