Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools
Yannick Guedes Bonthonneau, Thibault Lefeuvre · Annales de l’institut Fourier · 2022
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.