Revista Matemática Iberoamericana

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

Ilkka Holopainen , Steen Markvorsen , and Vicente Palmer

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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.

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Rev. Mat. Iberoamericana, Volume 25, Number 2 (2009), 709-738.

First available in Project Euclid: 13 October 2009

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Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25] 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 31C45: Other generalizations (nonlinear potential theory, etc.) 60J65: Brownian motion [See also 58J65]

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), no. 2, 709--738.

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