Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

TL;DR

This paper proves that noncompact normally hyperbolic invariant manifolds persist under perturbations in Riemannian manifolds with bounded geometry, emphasizing the importance of bounded geometry for uniform estimates.

Contribution

It establishes a persistence theorem for noncompact normally hyperbolic invariant manifolds specifically in manifolds of bounded geometry, extending classical results to noncompact settings.

Findings

Persistence of noncompact normally hyperbolic invariant manifolds proven

Bounded geometry is essential for uniform estimates in the proof

Results extend classical invariant manifold theory to noncompact manifolds

Abstract

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.