Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory

Yannick Guedes Bonthonneau; Thibault Lefeuvre

doi:10.5802/jep.248

Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory
[Rigidité locale des variétés à pointes hyperboliques II. Théorie non linéaire]

Yannick Guedes Bonthonneau ¹ ; Thibault Lefeuvre ²

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.

Résumé
Abstract

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].

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].

Reçu le : 2022-09-12
Accepté le : 2023-11-08
Publié le : 2023-11-16

DOI : 10.5802/jep.248

Classification : 53C24, 53C22, 37C27, 37D40
Keywords: Marked length spectrum, hyperbolic cusps, microlocal analysis, geometric rigidity
Mot clés : Spectre marqué des longueurs, pointes hyperboliques, analyse microlocale, rigidité géométrique

Affiliations des auteurs :

Yannick Guedes Bonthonneau ¹ ; Thibault Lefeuvre 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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Yannick Guedes Bonthonneau; Thibault Lefeuvre. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.248/

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