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Torsional rigidity of submanifolds with controlled geometry

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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-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.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Jaume I, 12071, Castelló, Spain

    A. Hurtado & V. Palmer

  2. Department of Mathematics, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    S. Markvorsen

Authors
  1. A. Hurtado
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  2. S. Markvorsen
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  3. V. Palmer
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Corresponding author

Correspondence to A. Hurtado.

Additional information

S. Markvorsen was supported by the Danish Natural Science Research Council and the Spanish MEC-DGI grant MTM2007-62344. V. Palmer was supported by Spanish MEC-DGI grant No. MTM2007-62344 and the Caixa Castelló Foundation and a grant of the Spanish MEC Programa de Estancias de profesores e investigadores españoles en centros de enseñanza superior e investigación extranjeros. A. Hurtado was supported by Spanish MEC-DGI grant No. MTM2007-62344, the Caixa Castelló Foundation.

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Hurtado, A., Markvorsen, S. & Palmer, V. Torsional rigidity of submanifolds with controlled geometry. Math. Ann. 344, 511–542 (2009). https://doi.org/10.1007/s00208-008-0315-3

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  • DOI: https://doi.org/10.1007/s00208-008-0315-3

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