Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity ofextrinsic domains of minimal submanifolds P^m in ambient Riemannianmanifolds Nⁿ with a pole p. The upper bounds are given in termsof the torsional rigidities of corresponding Schwarz symmetrizationsof the domains in warped product model spaces. Our main resultsare obtained via previously established isoperimetric inequalities,which are here extended to hold for this more general settingbased on warped product comparison spaces. We also characterizethe geometry of those situations in which the upper bounds forthe torsional rigidity are actually attained and give conditionsunder which the geometric average of the stochastic mean exittime for Brownian motion at infinity is finite. 2000 MathematicsSubject Classification 53C42, 58J65, 35J25, 60J65.