The inverse mean curvature flow in warped cylinders of non-positive radial curvature

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.