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 -) 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 -manifolds.
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 -opérateur pseudo-différentiel) lorsque le paramètre de la résolvante tend vers . 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 -variétés.
Accepted:
Published online:
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
Daniel Grieser 1; Mohammad Talebi 1; Boris Vertman 1
@article{JEP_2022__9__959_0, author = {Daniel Grieser and Mohammad Talebi and Boris Vertman}, title = {Spectral geometry on manifolds with fibered boundary metrics {I:} {Low} energy resolvent}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {959--1019}, publisher = {\'Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.198}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.198/} }
TY - JOUR AU - Daniel Grieser AU - Mohammad Talebi AU - Boris Vertman TI - Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 959 EP - 1019 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.198/ DO - 10.5802/jep.198 LA - en ID - JEP_2022__9__959_0 ER -
%0 Journal Article %A Daniel Grieser %A Mohammad Talebi %A Boris Vertman %T Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 959-1019 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.198/ %R 10.5802/jep.198 %G en %F JEP_2022__9__959_0
Daniel Grieser; Mohammad Talebi; Boris Vertman. Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 959-1019. doi : 10.5802/jep.198. https://jep.centre-mersenne.org/articles/10.5802/jep.198/
[ARS21] - Resolvent, heat kernel, and torsion under degeneration to fibered cusps, Mem. Amer. Math. Soc., vol. 269, no. 1314, American Mathematical Society, Providence, RI, 2021 | DOI
[BBLZ09] - “The Calderón projection: new definition and applications”, J. Geom. Phys. 59 (2009) no. 7, p. 784-826 | DOI | Zbl
[BGV04] - Heat kernels and Dirac operators, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 2004
[CCH06] - “Riesz transform and -cohomology for manifolds with Euclidean ends”, Duke Math. J. 133 (2006) no. 1, p. 59-93 | DOI | MR | Zbl
[FKS18] - “Monopoles and the Sen conjecture”, 2018 | arXiv
[GH08] - “Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I”, Math. Ann. 341 (2008) no. 4, p. 859-896 | DOI | Zbl
[GH09] - “Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 4, p. 1553-1610 | DOI | Numdam | Zbl
[GH09] - “Pseudodifferential operator calculus for generalized -rank 1 locally symmetric spaces. I”, J. Funct. Anal. 257 (2009) no. 12, p. 3748-3801 | DOI | MR | Zbl
[GH14] - “A parametrix construction for the Laplacian on -rank 1 locally symmetric spaces”, in Fourier analysis, Trends Math., Birkhäuser/Springer, Cham, 2014, p. 149-186 | DOI | Zbl
[Gri01] - “Basics of the -calculus”, in Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., vol. 125, Birkhäuser, Basel, 2001, p. 30-84 | DOI | MR | Zbl
[GS15] - “Low energy resolvent for the Hodge Laplacian: applications to Riesz transform, Sobolev estimates, and analytic torsion”, Internat. Math. Res. Notices (2015) no. 15, p. 6136-6210 | DOI | MR | Zbl
[HHM04] - “Hodge cohomology of gravitational instantons”, Duke Math. J. 122 (2004) no. 3, p. 485-548 | DOI | MR | Zbl
[HMM95] - “Analytic surgery and the accumulation of eigenvalues”, Comm. Anal. Geom. 3 (1995) no. 1-2, p. 115-222 | DOI | MR | Zbl
[KR22] - “Low energy limit for the resolvent of some fibered boundary operators”, Comm. Math. Phys. 390 (2022) no. 1, p. 231-307 | DOI | MR | Zbl
[KS15] - “Partial compactification of monopoles and metric asymptotics”, 2015 | arXiv
[Maz91] - “Elliptic theory of differential edge operators. I”, Comm. Partial Differential Equations 16 (1991) no. 10, p. 1615-1664 | DOI | MR | Zbl
[Mel92] - “Calculus of conormal distributions on manifolds with corners”, Internat. Math. Res. Notices (1992) no. 3, p. 51-61 | DOI | MR | Zbl
[Mel93] - The Atiyah-Patodi-Singer index theorem, Research Notes in Math., vol. 4, A K Peters, Ltd., Wellesley, MA, 1993 | DOI
[Mel96] - “Differential analysis on manifolds with corners”, 1996, http://www-math.mit.edu/~rbm/book.html
[Mel08] - “Real blow ups. Introduction to analysis on singular spaces”, 2008, Notes for lectures at MSRI, http://math.mit.edu/~rbm/InSisp/FrontI.pdf
[MM98] - “Pseudodifferential operators on manifolds with fibred boundaries”, Asian J. Math. 2 (1998) no. 4, p. 833-866 | DOI | MR | Zbl
[MSB]
, Unpublished notes[She13] - “The heat kernel on an asymptotically conic manifold”, Anal. PDE 6 (2013) no. 7, p. 1755-1791, Erratum: Ibid. 13 (2020), no. 3, p. 943–944 | DOI | MR | Zbl
[Vai01] - Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Math. Schriften, vol. 344, Universität Bonn, Mathematisches Institut, Bonn, 2001, arXiv:math/0102072 | MR
Cited by Sources: