How to obtain transience from bounded radial mean curvature

We show that Brownian motion on any unbounded submanifold in an ambient manifold with a pole is transient if the following conditions are satisfied: The -radial mean curvatures of are sufficiently small outside a compact set and the -radial sectional curvatures of are sufficiently negative. The `sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.