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Spectral Geometry and Asymptotically Conic Convergence Julie Marie Rowlett
 

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 k-form Laplacian under certain
geometric hypotheses. This theorem is proven using rescaling arguments, standard
elliptic techniques, and the b-calculus 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,

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics