Vicente Palmer

Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds

The relative volume growth of minimal submanifolds

P^m

Comparison theory of Lorentzian distance with applications to spacelike hypersurfaces

in ambient Riemannian manifolds

On the characterization of parabolicity and hyperbolicity of submanifolds

N^n

Mean Curvature and Volume of Extrinsic Spheres in Hypersurfaces of Real Space Forms

with a pole

Asymptotically Extrinsic Tamed Submanifolds

p

On Deciding Whether a Submanifold Is Parabolic or Hyperbolic Using Its Mean Curvature

. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.

Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds

The volume growth of certain well-defined subsets of minimal submanifolds in riemannian spaces are compared with the volume growth of balls and spheres in space forms of constant curvature.

Transience and capacity of minimal submanifolds

In this paper we summarize some comparison results for the Lorentzian distance function in spacetimes, with applications to the study of the geometric analysis of the Lorentzian distance on spacelike hypersurfaces. In particular, we will consider spacelike hypersufaces whose image under the immersion is bounded in the ambient spacetime and derive sharp estimates for the mean curvature of such hypersurfaces under appropriate hypotheses on the curvature of the ambient spacetime. The results in this paper are part of our recent work [1], where complete details and further related results may be found.

Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below

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.