  We obtain upper bounds for the isoperimetric quotients of extrinsic balls of
submanifolds in ambient spaces which have a lower bound on their radial
sectional curvatures. The submanifolds are themselves only assumed to have
lower bounds on the radial part of the mean curvature vector field and on the
radial part of the intrinsic unit normals at the boundaries of the extrinsic
spheres, respectively. In the same vein we also establish lower bounds on the
mean exit time for Brownian motion in the extrinsic balls. In those cases,
where we may extend our analysis to hold all the way to infinity, we apply a
capacity comparison technique to obtain a sufficient condition for the
submanifolds to be parabolic, i.e. a condition which will guarantee that any
Brownian particle, which is free to move around in the whole submanifold, is
bound to eventually revisit any given neighborhood of its starting point with
probability 1. The results of this paper are in a rough sense dual to similar
results obtained previously by the present authors in complementary settings
where we assume that the curvatures are bounded from above.
