Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory

TL;DR

This paper proves the local rigidity of hyperbolic cusp manifolds under nonlinear metric perturbations that decay at infinity, extending previous results from compact negatively-curved manifolds to non-compact cases.

Contribution

It extends the local rigidity results to manifolds with hyperbolic cusps for nonlinear perturbations, using advanced microlocal analysis of the X-ray transform.

Findings

Manifolds with hyperbolic cusps are locally rigid under certain nonlinear metric perturbations.

The generalized X-ray transform operator $t$ fits within the microlocal framework for cusp manifolds.

The proof combines linear theory with detailed analytic study of the X-ray transform.

Abstract

This article is the second in a series of two whose aim is to extend a recent result of Guillarmou-Lefeuvre [arXiv:1806.04218] 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 [arXiv:1907.01809] and on a careful analytic study of the generalized X-ray transform operator $\Pi_2$. In particular, we prove that the latter fits in the microlocal theory for cusp manifolds developed in [arXiv:1411.5083, arXiv:1712.07832, arXiv:1907.01809].