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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 obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above.
We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P with controlled radial mean curvature in ambient Riemannian manifolds 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 Schwarz… (More)
We give a set of sufficient and necessary conditions for parabolicity 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.
where χ(S) is the Euler characteristic of the surface, B r denotes the geodesic r-ball in Hn(b) and Vol(S 2∩Bb,n r ) Vol(B r ) is the volume growth of the domains S2 ∩B r . A natural question arises in this context: can we prove the finiteness of the topology of a not necessarily minimal surface in a Cartan–Hadamard manifold and, moreover, establish a… (More)