  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.
