 
Summary: Spectral Geometry and Asymptotically Conic Convergence
Julie Marie Rowlett
September 26, 2007
Abstract
In this paper we define asymptotically conic convergence in which a family of smooth
Riemannian metrics degenerates to have an isolated conic singularity. For a conic met
ric (M0, g0) and an asymptotically conic (scattering) metric (Z, gz) we define a non
standard blowup, the resolution blowup, in which the conic singularity in M0 is resolved
by Z. Equivalently, the resolution blowup resolves the boundary of the scattering met
ric using the conic metric; the resolution space is a smooth compact manifold. This
blowup induces a smooth family of metrics {g } on the compact resolution space M,
and we say (M, g ) converges asymptotically conically to (M0, g0) as 0.
Let and 0 be geometric Laplacians on (M, g ) and (M0, g0), respectively. Our
first result is convergence of the spectrum of to the spectrum of 0 as 0. Note
that this result implies spectral convergence for the kform Laplacian under certain
geometric hypotheses. This theorem is proven using rescaling arguments, standard
elliptic techniques, and the bcalculus of [26]. Our second result is technical: we con
struct a parameter ( ) dependent heat operator calculus which contains, and hence
describes precisely, the heat kernel for as 0. The consequences of this result
include: the existence of a polyhomogeneous asymptotic expansion for H as 0,
