We show that Brownian motion on any unbounded submanifold P in an ambient manifold N with a pole p is transient if the following conditions are satisfied: The p-radial mean curvatures of P are sufficiently small outside a compact set and the p-radial sectional curvatures of N are sufficiently negative. The 'sufficiency' conditions are obtained via… (More)
We give a set of sufficient and necessary conditions for parabol-icity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.
We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p ≥ 2.
We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P m with controlled radial mean curvature in ambient Riemannian manifolds N n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding… (More)