Normally hyperbolic invariant manifolds : the noncompact case

Abstract

In this work, we prove the persistence of normally hyperbolic invariant manifolds. This result is well known when the invariant manifold is compact; we extend this to a setting where the invariant manifold as well as the ambient space are allowed to be noncompact manifolds. The ambient space is assumed to be a Riemannian manifold of bounded geometry. Normally hyperbolic invariant manifolds (NHIMs) are a generalization of hyperbolic fixed points. Many of the concepts, results, and proofs for hyperbolic fixed points carry over to NHIMs. Two important properties that generalize to NHIMs are persistence of the invariant manifold and existence of stable and unstable manifolds.