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 ([R0,∞)×Sn,g¯) with metric g¯=dr2+ϑ2(r)σ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over Sn the flow exists for all times and remains a graph over Sn. 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 C2-estimates in case that the ambient pinching improves.