Revista Matemática Iberoamericana

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Volume 25, Issue 2, 2009, pp. 709–738
DOI: 10.4171/RMI/580

Published online: 2009-08-31

$p$-Capacity and $p$-Hyperbolicity of Submanifolds

Ilkka Holopainen[1], Steen Markvorsen[2] and Vicente Palmer[3]

(1) University of Helsinki, Finland
(2) Technical University of Denmark, Lyngby, Denmark
(3) Universitat Jaume I, Castellón, Spain

We use explicit solutions to a drifted Laplace equation in warped product model spaces as comparison constructions to show $p$-hyperbolicity of a large class of submanifolds for $p\ge 2$. The condition for $p$-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the radial convexity of the submanifold. In the process of showing $p$-hyperbolicity we also obtain explicit lower bounds on the $p$-capacity of finite annular domains of the submanifolds in terms of the drifted $2$-capacity of the corresponding annuli in the respective comparison spaces.

Keywords: Submanifolds, transience, p-Laplacian, hyperbolicity, parabolicity, capacity, isoperimetric inequality, comparison theory

Holopainen Ilkka, Markvorsen Steen, Palmer Vicente: $p$-Capacity and $p$-Hyperbolicity of Submanifolds. Rev. Mat. Iberoamericana 25 (2009), 709-738. doi: 10.4171/RMI/580