  We consider the inverse mean curvature flow in smooth Riemannian manifolds of
the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric
$\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional
curvature. We prove, that for initial mean-convex graphs over $S^n$ the flow
exists for all times and remains a graph over $S^{n}$. Under weak further
assumptions on the ambient manifold, we prove optimal decay of the gradient and
that the flow leaves become umbilic exponentially fast. We prove optimal $C^2$
estimates in case that the ambient pinching improves.
