Torsional rigidity of submanifolds with controlled geometry

Abstract

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds Pm with controlled radial mean curvature in ambient Riemannian manifolds Nn with a pole p and with sectional curvatures bounded from above and from below, respec- tively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped prod- uct 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 situ- ations in which the bounds for the torsional rigidity are actually attained and study the behavior at in¯nity of the so-called geometric average of the mean exit time for Brownian motion