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

Grieser, Daniel; Talebi, Mohammad; Vertman, Boris

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 959-1019.

Nous étudions la résolvante à basse énergie du laplacien de Hodge sur une variété munie d’une métrique fibrée au bord. Nous déterminons le comportement asymptotique précis de la résolvante en tant qu’opérateur pseudo-différentiel fibré au bord (aussi appelé $\varphi$-opérateur pseudo-différentiel) lorsque le paramètre de la résolvante tend vers $0$. Ceci généralise les travaux précédents de Guillarmou et Sher qui considéraient les métriques asymptotiquement coniques, correspondant au cas particulier où les fibres sont des points. Le phénomène nouveau dans le cas de fibres non triviales est que la résolvante a un comportement asymptotique différent sur le sous-espace des formes qui sont harmoniques dans la direction des fibres et sur son supplémentaire orthogonal. Pour traiter ce problème, nous introduisons un calcul pseudo-différentiel « décomposé » approprié, en nous appuyant sur les travaux de Grieser et Hunsicker et en les étendant. Notre travail jette les bases d’une discussion des invariants spectraux sur les $\varphi$-variétés.

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibered boundary (aka $\varphi$-) 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 fibers are points. The new feature in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise 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 $\varphi$-manifolds.

Classification : 58J05, 58J40, 35J70
Keywords: Low energy resolvent, pseudodifferential calculus, scattering metric, manifolds with fibered boundary
Mot clés : Résolvante à basse énergie, calcul pseudo-différentiel, métrique de diffusion, variété avec métrique fibrée au bord

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