Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

TL;DR

This paper proves the persistence of noncompact normally hyperbolic invariant manifolds within Riemannian manifolds of bounded geometry, establishing optimal smoothness and new foundational results for such submanifolds.

Contribution

It introduces a persistence theorem for noncompact invariant manifolds in bounded geometry and develops new uniform approximation and tubular neighborhood results.

Findings

Persistence of noncompact invariant manifolds established

New uniform tubular neighborhood theorem derived

Optimal $C^{k,eta}$ smoothness result proven

Abstract

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,\alpha}$-smoothness result is optimal with respect to the spectral gap condition involved. The core of the persistence proof is based on the Perron method. In the process we derive new results on noncompact submanifolds in bounded geometry: a uniform tubular neighborhood theorem and uniform smooth approximation of a submanifold. The submanifolds considered are assumed to be uniformly $C^k$ bounded in an appropriate sense.