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Abstract. 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.

We state and prove a Chern-Osserman-type Inequality in terms of the volume growth for minimal surfaces S which have finite total extrinsic curvature and are properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0 and such that they are not too curved (on average) with respect to the Hyperbolic space with constant sectional curvature given by the upper bound b. We have also proven the same Chern-Osserman-type Inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0.2000 Mathematics Subject Classification. Primary 53A15, 53C20; Secondary 53C42.is the geodesic t-ball centered at the pole o in the ambient space N , and B b,2 t denotes the geodesic t-ball in H 2 (b). Remark 1.2. The main theorem in [7] is a corollary of Theorem 1.1. In fact, note that condition (1.4) is superfluous when the ambient manifold is H n (b). On the other hand, when the ambient manifold is H n (b), then condition (1.3) implies that A S (q) goes

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