Torsional rigidity of submanifolds with controlled geometry

TL;DR

This paper establishes explicit bounds for the torsional rigidity of submanifolds with controlled geometry in Riemannian manifolds, using symmetrization and isoperimetric inequalities, and characterizes cases of equality and asymptotic behavior.

Contribution

It provides new explicit bounds for torsional rigidity of submanifolds with controlled radial mean curvature in ambient spaces, extending previous isoperimetric inequality methods.

Findings

Derived explicit upper and lower bounds for torsional rigidity.

Characterized geometric conditions for bounds to be attained.

Analyzed asymptotic behavior of mean exit time for Brownian motion.

Abstract

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 Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in e.g. [MP4] and [MP5]. As in [MP4] we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion.