Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Abstract

We prove a persistence result for noncompact normally hyperbolic invariant manifolds (NHIMs) in the setting of Riemannian manifolds of bounded geometry. This generalizes classical results by Fenichel and by Hirsch, Pugh and Shub from compact to noncompact manifolds. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. Normally hyperbolic invariant manifolds (NHIMs) are generalizations of hyperbolic fixed points. Their properties are similar to center manifolds, but they are globally uniquely defined. We prove that a noncompact NHIM M persist under C^1 small perturbations of the vector field and that the persistent manifold is C^r when M is an r-NHIM, where r = k + \alpha may include a Hölder continuity condition. This C^{k,\alpha}-smoothness result is optimal with respect to the spectral gap condition involved. The ambient manifold must have bounded geometry, that is, its curvature and injectivity radius must be globally bounded. As a result, classes of uniformly continuous C^k bounded functions, vector fields and submanifolds can be defined. The NHIM is assumed to be a non-injectively immersed (possibly noncompact) submanifold in this setting. The core of the persistence proof is formulated in a trivial bundle setting and based on the Perron method. We use ideas of Henry to generalize the Perron method to NHIMs, and the fiber contraction theorem and ideas of Vanderbauwhede and Van Gils to prove higher order smoothness. In the process we require bounded curvature to estimate certain holonomy terms. This result is then generalized to the setting of an ambient manifold of bounded geometry. To this end, we derive new results on noncompact submanifolds in bounded geometry: a uniform tubular neighborhood theorem, uniform smooth approximation of a submanifold and a theorem on embedding a normal bundle into a trivial bundle. The submanifolds considered are assumed to be uniformly C^k bounded in an appropriate sense and may be non-injectively immersed. Finally, some extensions of the main result are discussed. Time-dependent systems, parameter dependence and overflowing invariant manifolds and a small generalization thereof