Conic degeneration and the determinant of the Laplacian

Abstract

We investigate the behavior of various spectral invariants, particularly the determinant of the Laplacian, on a family of smooth Riemannian manifolds Ω ge that undergo conic degeneration, i.e., that converge in a particular way to a manifold with a conical singularity. Our main result is an asymptotic formula for the determinant up to terms that vanish as ∈ goes to 0. The proof proceeds in two parts: we study the fine structure of the heat trace on the degenerating manifolds via a parametrix construction, and then use that fine structure to analyze the zeta function and determinant of the Laplacian.