Spectral geometry and asymptotically conic convergence

Abstract

We define a new conic metric collapse, asymptotically conic convergence,1 in which a family of smooth Riemannian metrics degenerates to have an isolated conic singularity. For a conic metric (Mo, go) and an asymptotically conic or "scattering" metric (Z, gz), we construct a new non-standard blowup, the resolution blowup. in which the conic singularity in Mo is resolved by Z. This blowup induces a smooth family of metrics {gε} on the compact resolution space M. (M, g ε) is said to converge asymptotically conically to (M o, go) as ε → 0. Let Δε and Δ0 be geometric Laplacians on (M, g0) and (Mo, go), respectively. Our first result is convergence of the spectrum of Δε to the spectrum of Δ0 as ε → 0. Note that this result implies spectral convergence for the k-form Laplacian under certain geometric hypotheses. This theorem is proven using rescating arguments, standard elliptic techniques and the b-calculus of [33]. Our second result is technical: we construct a parameter (ε) dependent heat operator calculus which contains, and hence describes precisely, the heat kernel for Δ as ε → 0. The consequences of this result include the existence of a polyhomogeneous asymptotic expansion for Hε as ε → 0, with uniform convergence down to t = 0. To prove this result, we construct heat spaces as manifolds with corners using both standard and non-standard blowups on which we construct suitable heat operator calculi. A parametrix construction modeled after Melrose's heat kernel construction [33] and a maximum principle argument complete the proof.