Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent

Abstract

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka φ-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on φ-manifolds.

Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

“…Outlook and upcoming work. In the upcoming work by the first author, the presented heat kernel asymptotics together with the low energy resolvent, as constructed in our previous work [GTV20] jointly with Daniel Grieser, is applied to define the renormalized heat trace and study its asymptotics. This leads to the definition of a renormalized analytic torsion for φ-manifolds.…”

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In this paper we continue with the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as φ-metrics, as initiated in our previous work [GTV20]. We consider the Hodge Laplacian for a φ-metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners.

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A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

“…In particular, the basic construction of the blowup along a p-submanifold coincides with the one in these references. However, the lifting (or pullback) of submanifolds differs in certain cases from the one used in, for instance, [30,37,42]. As a consequence, our notion of iterated blow-up exhibits some subtle differences to the iterated blow-up discussed in the aforementioned papers.…”

“…We have t ≥ 1 if and only if s ≤ 2/(n − 2), and in this case the Coulomb assumption implies the boundedness of ρ t F V . Then, regularity theory yields for any η ∈ C ∞ (X F ): if û ∈ ηL 2 (X F , W F ) ∩ η 1/(s+1) L 2/(s+1) (X F , W F ) (30) then û ∈ ηW 2,2 (X F , W F ). (31) At first, we consider the linear map F A,V : n+1 → n+1 , p) → (t, Ap + t V ).…”

Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

“…In the upcoming work by the first author, the presented heat kernel asymptotics together with the low energy resolvent, as constructed in our previous work [8] jointly with Daniel Grieser, is applied to define the renormalized heat trace and study its asymptotics. This leads to the definition of a renormalized analytic torsion for φ-manifolds.…”