Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory
[Rigidité locale des variétés à pointes hyperboliques II. Théorie non linéaire]
Guedes Bonthonneau, Yannick1 ; Lefeuvre, Thibault2
1 Université Paris Nord, CNRS, LAGA Villetaneuse, France
2 Université de Paris and Sorbonne Université, CNRS, IMJ-PRG F-75006 Paris, France.
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1441-1510

This article is the second in a series of two whose aim is to extend a recent result of Guillarmou-Lefeuvre [GL19] 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. We deal with the nonlinear version of the problem and prove that such manifolds are locally rigid for nonlinear perturbations of the metric that slightly decrease at infinity. Our proof relies on the linear theory addressed in [GBL23a] and on a careful analytic study of the generalized X-ray transform operator Π 2 . In particular, we prove that the latter fits in the microlocal theory for cusp manifolds developed in [GB16, GBW22, GBL23a].

Cet article est le second d’une série de deux visant à étendre un résultat récent de Guillarmou-Lefeuvre [GL19] sur la rigidité locale du spectre des longueurs marquées, passant du cas des variétés riemanniennes compactes à courbure négative au cas des variétés à pointes hyperboliques. Nous abordons la version non linéaire du problème et montrons que de telles variétés sont localement rigides pour des perturbations non linéaires de la métrique qui décroissent légèrement à l’infini. Notre démonstration repose sur la théorie linéaire abordée dans [GBL23a] et sur une étude analytique approfondie de l’opérateur de transformée en rayons X généralisée Π 2 . En particulier, nous montrons que ce dernier s’inscrit dans la théorie microlocale des variétés à pointes développée dans [GB16, GBW22, GBL23a].

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.248
Classification : 53C24, 53C22, 37C27, 37D40
Keywords: Marked length spectrum, hyperbolic cusps, microlocal analysis, geometric rigidity
Mots-clés : Spectre marqué des longueurs, pointes hyperboliques, analyse microlocale, rigidité géométrique

Guedes Bonthonneau, Yannick 1 ; Lefeuvre, Thibault 2

1 Université Paris Nord, CNRS, LAGA Villetaneuse, France
2 Université de Paris and Sorbonne Université, CNRS, IMJ-PRG F-75006 Paris, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Guedes Bonthonneau, Yannick; Lefeuvre, Thibault. Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1441-1510. doi: 10.5802/jep.248

[AB22] Adam, Alexander; Baladi, Viviane Horocycle averages on closed manifolds and transfer operators, Tunis. J. Math., Volume 4 (2022) no. 3, pp. 387-441 | DOI | MR | Zbl

[BCD11] Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wissen., 343, Springer, Berlin-Heidelberg, 2011 | DOI

[BCG95] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Volume 5 (1995) no. 5, pp. 731-799 | DOI

[BK85] Burns, Keith; Katok, Anatole Manifolds with nonpositive curvature, Ergodic Theory Dynam. Systems, Volume 5 (1985) no. 2, pp. 307-317 | DOI | MR

[BT08] Baladi, Viviane; Tsujii, Masato Dynamical determinants and spectrum for hyperbolic diffemorphisms, Geometric and probabilistic structures in dynamics (Contemp. Math), Volume 469, American Mathematical Society, Providence, RI, 2008, pp. 29-68 | DOI | Zbl

[Cao95] Cao, Jian Guo Rigidity for non-compact surfaces of finite area and certain Kähler manifolds, Ergodic Theory Dynam. Systems, Volume 15 (1995) no. 3, pp. 475-516 | DOI | MR

[CL22] Cekić, Mihajlo; Lefeuvre, Thibault Generic dynamical properties of connections on vector bundles, Internat. Math. Res. Notices (2022) no. 14, pp. 10649-10703 | DOI | MR | Zbl

[Con92] Contreras, Gonzalo Regularity of topological and metric entropy of hyperbolic flows, Math. Z., Volume 210 (1992) no. 1, pp. 97-112 | DOI | MR | Zbl

[Cro90] Croke, Christopher B. Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 150-169 | DOI | MR | Zbl

[Cro04] Croke, Christopher B. Rigidity theorems in Riemannian geometry, Geometric methods in inverse problems and PDE control (IMA Vol. Math. Appl.), Volume 137, Springer, New York, 2004, pp. 47-72 | DOI | MR | Zbl

[DGRS20] Dang, Nguyen Viet; Guillarmou, Colin; Rivière, Gabriel; Shen, Shu The Fried conjecture in small dimensions, Invent. Math., Volume 220 (2020) no. 2, pp. 525-579 | DOI | MR | Zbl

[dlLMM86] de la Llave, Rafael; Marco, José Manuel; Moriyón, Roberto Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2), Volume 123 (1986) no. 3, pp. 537-611 | DOI | Zbl

[dlLO99] de la Llave, Rafael; Obaya, Rafael Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, Volume 5 (1999) no. 1, pp. 157-184 | DOI | Zbl

[DZ16] Dyatlov, Semyon; Zworski, Maciej Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. École Norm. Sup. (4), Volume 49 (2016) no. 3, pp. 543-577 | DOI | MR | Zbl

[Ebi68] Ebin, David G. On the space of Riemannian metrics, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 1001-1003 | DOI | MR | Zbl

[FS11] Faure, Frédéric; Sjöstrand, Johannes Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., Volume 308 (2011) no. 2, pp. 325-364 | DOI | MR | Zbl

[FT13] Faure, Frédéric; Tsujii, Masato Band structure of the Ruelle spectrum of contact Anosov flows, Comptes Rendus Mathématique, Volume 351 (2013) no. 9-10, pp. 385-391 | DOI | MR | Numdam | Zbl

[GB15] Guedes Bonthonneau, Yannick Résonances du laplacien sur les variétés à pointes, Ph. D. Thesis, Université Paris Sud XI (2015) (https://theses.hal.science/tel-01194752)

[GB16] Guedes Bonthonneau, Yannick Long time quantum evolution of observables on cusp manifolds, Comm. Math. Phys., Volume 343 (2016) no. 1, pp. 311-359 | DOI | MR | Zbl

[GBL23a] Guedes Bonthonneau, Yannick; Lefeuvre, Thibault Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools, Ann. Inst. Fourier (Grenoble), Volume 73 (2023) no. 1, pp. 335-421 | DOI | MR | Zbl

[GBL23b] Guedes Bonthonneau, Yannick; Lefeuvre, Thibault Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics, Ann. H. Lebesgue, Volume 6 (2023), pp. 643-686 | DOI | Zbl

[GBW22] Guedes Bonthonneau, Yannick; Weich, Tobias Ruelle-Pollicott resonances for manifolds with hyperbolic cusps, J. Eur. Math. Soc. (JEMS), Volume 24 (2022) no. 3, pp. 851-923 | DOI | MR | Zbl

[GKL22] Guillarmou, Colin; Knieper, Gerhard; Lefeuvre, Thibault Geodesic stretch, pressure metric and marked length spectrum rigidity, Ergodic Theory Dynam. Systems, Volume 42 (2022) no. 3, p. 974–1022 | DOI | MR | Zbl

[GL19] Guillarmou, Colin; Lefeuvre, Thibault The marked length spectrum of Anosov manifolds, Ann. of Math. (2), Volume 190 (2019) no. 1, pp. 321-344 | MR | Zbl | DOI

[GL21] Gouëzel, Sébastien; Lefeuvre, Thibault Classical and microlocal analysis of the x-ray transform on Anosov manifolds, Anal. PDE, Volume 14 (2021) no. 1, pp. 301-322 | DOI | MR | Zbl

[GLP13] Giulietti, Paolo; Liverani, Carlangelo; Pollicott, Mark Anosov flows and dynamical zeta functions, Ann. of Math. (2), Volume 178 (2013) no. 2, pp. 687-773 | DOI | MR | Zbl

[Gre56] Green, Leon W. Geodesic Instability, Proc. Amer. Math. Soc., Volume 7 (1956) no. 3, pp. 438-448 | MR | Zbl | DOI

[Gro00] Gromov, Mikhaïl Three remarks on geodesic dynamics and fundamental group, Enseign. Math. (2), Volume 46 (2000) no. 3-4, pp. 391-402 | MR | Zbl

[GS94] Grigis, Alain; Sjöstrand, Johannes Microlocal analysis for differential operators, London Math. Society Lect. Note Series, 196, Cambridge University Press, Cambridge, 1994 | DOI

[Gui17] Guillarmou, Colin Invariant distributions and X-ray transform for Anosov flows, J. Differential Geom., Volume 105 (2017) no. 2, pp. 177-208 http://projecteuclid.org/euclid.jdg/1486522813 | MR | Zbl

[Ham99] Hamenstädt, Ursula Cocycles, symplectic structures and intersection, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 90-140 | DOI | MR | Zbl

[Has92] Hasselblatt, Boris Bootstrapping regularity of the Anosov splitting, Proc. Amer. Math. Soc., Volume 115 (1992) no. 3, pp. 817-819 | DOI | MR | Zbl

[Kat88] Katok, Anatole Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems, Volume 8 * (1988), pp. 139-152 (Charles Conley Memorial Issue) | DOI | MR | Zbl

[Kat95] Kato, Tosio Perturbation theory for linear operators, Class. Math., Springer-Verlag, Berlin, 1995 | DOI

[KKPW89] Katok, A.; Knieper, G.; Pollicott, M.; Weiss, H. Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., Volume 98 (1989) no. 3, pp. 581-597 | DOI | MR | Zbl

[Kli74] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl

[Kni02] Knieper, Gerhard Hyperbolic dynamical systems, Handbook of dynamical systems (Hasselblatt, B.; Katok, A., eds.), Volume 1A, Elsevier, Amsterdam, 2002, pp. 239-319

[Lef19] Lefeuvre, Thibault Sur la rigidité des variétés riemanniennes, Ph. D. Thesis, Université Paris-Saclay (2019) (https://theses.hal.science/tel-02462364)

[Liv72] Livšic, A. N. Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., Volume 36 (1972), pp. 1296-1320 | MR

[Mel93] Melrose, Richard B. The Atiyah-Patodi-Singer index theorem, Research Notes in Math., 4, A K Peters, Ltd., Wellesley, MA, 1993 | DOI

[NT98] Niţică, Viorel; Török, Andrei Regularity of the transfer map for cohomologous cocycles, Ergodic Theory Dynam. Systems, Volume 18 (1998) no. 5, pp. 1187-1209 | DOI | MR | Zbl

[Ota90] Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2), Volume 131 (1990) no. 1, pp. 151-162 | DOI | Zbl

[Pat99] Paternain, Gabriel P. Geodesic flows, Progress in Math., 180, Birkhäuser Boston, Inc., Boston, MA, 1999 | DOI

[PPS15] Paulin, Frédéric; Pollicott, Mark; Schapira, Barbara Equilibrium states in negative curvature, Astérisque, 373, Société Mathématique de France, Paris, 2015

[Riq18] Riquelme, Felipe Ruelle’s inequality in negative curvature, Discrete Contin. Dynam. Systems, Volume 38 (2018) no. 6, pp. 2809-2825 | DOI | MR | Zbl

[Shu01] Shubin, Mikhail A. Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 2001 | DOI

[Tay97] Taylor, Michael E. Partial differential equations. III, Applied Math. Sciences, 117, Springer-Verlag, New York, 1997

[Wil14] Wilkinson, Amie Lectures on marked length spectrum rigidity, Geometric group theory (IAS/Park City Math. Ser.), Volume 21, American Mathematical Society, Providence, RI, 2014, pp. 283-324 | DOI | MR | Zbl

[Zwo12] Zworski, Maciej Semiclassical analysis, Graduate Studies in Math., 138, American Mathematical Society, Providence, RI, 2012 | DOI

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