Torsional rigidity of submanifolds with controlled geometry

Ana Hurtado, Steen Markvorsen, Vicente Palmer

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    Abstract

    We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds ^m$ with controlled radial mean curvature in ambient Riemannian manifolds ^n$ with a pole $ 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.
    Original languageEnglish
    JournalMathematische Annalen
    Volume344
    Issue number3
    Pages (from-to)511-542
    ISSN0025-5831
    DOIs
    Publication statusPublished - 2009

    Keywords

    • Schwarz symmetrization
    • Torsional rigidity
    • Submanifolds

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