[Rigidité locale des variétés à pointes hyperboliques I. Théorie linéaire et outils microlocaux.]
Cet article est le premier d’une série de deux, dont le but est d’étendre un récent résultat de Guillarmou et du second auteur sur la rigidité locale du spectre marqué des longueurs, du cas des variétés riemanniennes compactes à courbure négative au cas des variétés à pointes hyperboliques. Dans ce premier article, nous traitons la version linéaire (ou infinitésimale) du problème et prouvons que de telles variétés sont spectralement rigides pour des déformations à support compact. Plus précisément, nous prouvons que la transformée en rayons X sur les -tenseurs symétriques solénoïdaux est injective. Pour ce faire, nous développons un calcul microlocal introduit par Bonthonneau et Bonthonneau–Weich, nous permettant d’inverser des opérateurs pseudodifférentiels sur des espaces de Sobolev et de Hölder–Zygmund, modulo des restes compacts. Cette théorie, qui a un intérêt en soi, sera utilisée de façon cruciale dans notre second article pour traiter le problème non linéaire.
This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou and of the second author on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. In this first paper, we deal with the linear (or infinitesimal) version of the problem and prove that such manifolds are spectrally rigid for compactly supported deformations. More precisely, we prove that the X-ray transform on symmetric solenoidal 2-tensors is injective. In order to do so, we expand the microlocal calculus developed by Bonthonneau and Bonthonneau–Weich to be able to invert pseudo-differential operators on Sobolev and Hölder–Zygmund spaces modulo compact remainders. This theory has an interest on its own and will be extensively used in the second paper in order to deal with the nonlinear problem.
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Keywords: Marked length spectrum, Inverse problem, Microlocal analysis, Non-compact manifolds
Mot clés : Spectre marqué des longueurs, Problème inverse, Analyse microlocale, Variétés non-compactes
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@article{AIF_2023__73_1_335_0,
author = {Guedes Bonthonneau, Yannick and Lefeuvre, Thibault},
title = {Local rigidity of manifolds with hyperbolic cusps {I.} {Linear} theory and microlocal tools},
journal = {Annales de l'Institut Fourier},
pages = {335--421},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {73},
number = {1},
year = {2023},
doi = {10.5802/aif.3534},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3534/}
}
TY - JOUR AU - Guedes Bonthonneau, Yannick AU - Lefeuvre, Thibault TI - Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools JO - Annales de l'Institut Fourier PY - 2023 SP - 335 EP - 421 VL - 73 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3534/ DO - 10.5802/aif.3534 LA - en ID - AIF_2023__73_1_335_0 ER -
%0 Journal Article %A Guedes Bonthonneau, Yannick %A Lefeuvre, Thibault %T Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools %J Annales de l'Institut Fourier %D 2023 %P 335-421 %V 73 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3534/ %R 10.5802/aif.3534 %G en %F AIF_2023__73_1_335_0
Guedes Bonthonneau, Yannick; Lefeuvre, Thibault. Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools. Annales de l'Institut Fourier, Tome 73 (2023) no. 1, pp. 335-421. doi : 10.5802/aif.3534. https://aif.centre-mersenne.org/articles/10.5802/aif.3534/
[1] Index theorems on manifolds with straight ends, Compos. Math., Volume 148 (2012) no. 6, pp. 1897-1968 | DOI | MR | Zbl
[2] Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995) no. 5, pp. 731-799 | DOI | MR | Zbl
[3] Résonances du laplacien sur les variétés à pointes, Ph. D. Thesis, Université Paris Sud - Paris XI, France (2015) (https://tel.archives-ouvertes.fr/tel-01194752)
[4] Weyl laws for manifolds with hyperbolic cusps (2015) (http://arxiv.org/abs/1512.05794)
[5] Long time quantum evolution of observables on cusp manifolds, Commun. Math. Phys., Volume 343 (2016) no. 1, pp. 311-359 | DOI | MR | Zbl
[6] Manifolds with nonpositive curvature, Ergodic Theory Dyn. Syst., Volume 5 (1985) no. 2, pp. 307-317 | DOI | MR | Zbl
[7] Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 150-169 | DOI | MR | Zbl
[8] Spectral rigidity of a compact negatively curved manifold, Topology, Volume 37 (1998) no. 6, pp. 1265-1273 | DOI | MR | Zbl
[9] The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | DOI | MR | Zbl
[10] Almost flat manifolds, J. Differ. Geom., Volume 13 (1978) no. 2, pp. 231-241 | DOI | MR | Zbl
[11] Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory (2019) (https://arxiv.org/abs/1910.02154)
[12] Ruelle–Pollicott Resonances for Manifolds with Hyperbolic Cusps, J. Eur. Math. Soc. (JEMS), Volume 24 (2022) no. 3, pp. 851-923 | DOI | MR | Zbl
[13] Invariant distributions and X-ray transform for Anosov flows, J. Differ. Geom., Volume 105 (2017) no. 2, pp. 177-208 | MR | Zbl
[14] The marked length spectrum of Anosov manifolds, Ann. Math., Volume 190 (2019) no. 1, pp. 321-344 | DOI | MR | Zbl
[15] Some inverse spectral results for negatively curved -manifolds, Topology, Volume 19 (1980) no. 3, pp. 301-312 | DOI | MR | Zbl
[16] Some inverse spectral results for negatively curved -manifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proceedings of Symposia in Pure Mathematics), Volume 36, American Mathematical Society (1980), pp. 153-180 | MR | Zbl
[17] Cocycles, symplectic structures and intersection, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 90-140 | DOI | MR | Zbl
[18] Hyperbolic flows (*) (In preparation)
[19] Can one hear the shape of a drum?, Am. Math. Mon., Volume 73 (1966) no. 4, part II, pp. 1-23 | DOI | MR | Zbl
[20] Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dyn. Syst., Volume 8 (1988) no. Charles Conley Memorial Issue, pp. 139-152 | DOI | MR | Zbl
[21] Riemannian geometry, De Gruyter Studies in Mathematics, 1, Walter de Gruyter, 1995 | DOI | MR | Zbl
[22] Scattering theory for automorphic functions, Annals of Mathematics Studies, 87, Princeton University Press, 1976 | MR | Zbl
[23] On the rigidity of Riemannian manifolds, 2019 (https://thibaultlefeuvre.files.wordpress.com/2019/12/slides-wout-pause-finale.pdf)
[24] Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math., Volume 2 (1998) no. 4, pp. 833-866 (Mikio Sato: a great Japanese mathematician of the twentieth century) | DOI | MR | Zbl
[25] The Atiyah–Patodi–Singer index theorem, Research Notes in Mathematics, 4, A K Peters, 1993 | DOI | MR | Zbl
[26] Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr., Volume 111 (1983), pp. 197-288 | DOI | MR | Zbl
[27] Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math., Volume 109 (1992) no. 2, pp. 265-305 | DOI | MR | Zbl
[28] Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math., Volume 131 (1990) no. 1, pp. 151-162 | DOI | MR | Zbl
[29] Geodesic flows, Progress in Mathematics, 180, Birkhäuser, 1999 | DOI | MR | Zbl
[30] Spectral rigidity and invariant distributions on Anosov surfaces, J. Differ. Geom., Volume 98 (2014) no. 1, pp. 147-181 | MR | Zbl
[31] Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Math. Ann., Volume 363 (2015) no. 1-2, pp. 305-362 | DOI | MR | Zbl
[32] Equilibrium states in negative curvature, Astérisque, 373, Société Mathématique de France, 2015 | MR | Zbl
[33] Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, Walter De Gruyter, 1994, 271 pages | DOI | MR | Zbl
[34] Pseudodifferential operators and spectral theory, Springer, 2001 (Translated from the 1978 Russian original by Stig I. Andersson) | DOI | MR | Zbl
[35] Complex scaling and the distribution of scattering poles, J. Am. Math. Soc., Volume 4 (1991) no. 4, pp. 729-769 | DOI | MR | Zbl
[36] Mapping properties of pseudodifferential operators on manifolds with bounded geometry, J. Lond. Math. Soc., Volume 57 (1998) no. 3, pp. 721-738 | DOI | MR | Zbl
[37] Partial differential equations. III: Nonlinear equations, Applied Mathematical Sciences, 117, Springer, 1996 (Nonlinear equations, Corrected reprint of the 1996 original) | DOI | MR | Zbl
[38] Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Mathematische Schriften [Bonn Mathematical Publications], 344, Universität Bonn, Mathematisches Institut, Bonn, 2001 (Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2001) | MR | Zbl
[39] Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compos. Math., Volume 27 (1973), p. 83-106, 159–184 | MR | Zbl
[40] Variétés riemanniennes isospectrales et non isométriques, Ann. Math., Volume 112 (1980) no. 1, pp. 21-32 | DOI | MR | Zbl
[41] Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | DOI | MR | Zbl
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