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This article is the second in a series of two whose aim is to extend a recentresult of Guillarmou-Lefeuvre [arXiv:1806.04218] on the local rigidity of themarked length spectrum from the case of compact negatively-curved Riemannianmanifolds to the case of manifolds with hyperbolic cusps. We deal with thenonlinear version of the problem and prove that such manifolds are locallyrigid for nonlinear perturbations of the metric that slightly decrease atinfinity. 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 theoryfor cusp manifolds developed in [arXiv:1411.5083, arXiv:1712.07832,arXiv:1907.01809].

Local rigidity of manifolds with hyperbolic cusps II. Nonlinear theory