Spectral geometry and asymptotically conic convergence

We define a new conic metric collapse, {\em asymptoticallyconic conver\-gence},$^1$ in which a family of smoothRiemannian metrics degenerates to have an isolated conicsingularity. For a conic metric $(M_0, g_0)$ and anasymptotically conic or “scattering” metric $(Z, g_z)$,we construct a new non-standard blowup, the \em resolutionblowup, \em in which the conic singularity in $M_0$ isresolved by $Z.$ This blowup induces a smooth family ofmetrics $\{ g_{\epsilon} \}$ on the compact resolutionspace $M.$ $(M, g_{\epsilon})$ is said to converge \emasymptotically conically \em to $(M_0 , g_0)$ as $\epsilon\to 0$.

Let $\Delta_{\epsilon}$ and $\Delta_0$ be geometricLaplacians on $(M, g_{\epsilon})$ and $(M_0, g_0),$respectively. Our first result is convergence of thespectrum of $\Delta_{\epsilon}$ to the spectrum of$\Delta_0$ as $\epsilon \to 0.$ Note that this resultimplies spectral convergence for the $k$-form Laplacianunder certain geometric hypotheses. This theorem is provenusing rescaling arguments, standard elliptic techniques andthe $b$-calculus of \cite{tapsit}. Our second result istechnical: we construct a parameter ($\epsilon$) dependentheat operator calculus which contains, and hence describesprecisely, the heat kernel for $\Delta_{\epsilon}$ as$\epsilon \to 0.$ The consequences of this result includethe existence of a polyhomogeneous asymptotic expansion for$H_{\epsilon}$ as $\epsilon \to 0,$ with uniformconvergence down to $t=0.$ To prove this result, weconstruct heat spaces as manifolds with corners using bothstandard and non-standard blowups on which we constructsuitable heat operator calculi. A parametrix constructionmodeled after Melrose’s heat kernel construction\cite{tapsit} and a maximum principle argument complete theproof.

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Published 1 January 2008