Mathematische Annalen

, Volume 344, Issue 3, pp 511–542 | Cite as

Torsional rigidity of submanifolds with controlled geometry

Article

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, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization 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., Markvorsen and Palmer (Proc Lond Math Soc 93:253--272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper 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.

Mathematics Subject Classification (2000)

Primary 53C42 58J65 35J25 60J65 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Jaume ICastellóSpain
  2. 2.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark

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