Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools
Annales de l'Institut Fourier, Online first, 87 p.

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.

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 2-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.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3534
Classification: 53C24,  53C22,  37C27,  37D40
Keywords: Marked length spectrum, Inverse problem, Microlocal analysis, Non-compact manifolds
Guedes Bonthonneau, Yannick 1; Lefeuvre, Thibault 2

1 Laga - Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse (France)
2 Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris (France)
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Guedes Bonthonneau, Yannick; Lefeuvre, Thibault. Local rigidity of manifolds with hyperbolic cusps  I. Linear theory and microlocal tools. Annales de l'Institut Fourier, Online first, 87 p.

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