Revista Matemática Iberoamericana

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

Ilkka Holopainen , Steen Markvorsen , and Vicente Palmer

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 13 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1255440072

Mathematical Reviews number (MathSciNet)
MR2569551

Zentralblatt MATH identifier
1176.53056

Subjects
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]

Keywords
submanifolds transience $p$-Laplacian hyperbolicity parabolicity capacity isoperimetric inequality comparison theory

Citation

Holopainen , Ilkka; Markvorsen , Steen; Palmer , Vicente. $p$-Capacity and $p$-Hyperbolicity of Submanifolds. Rev. Mat. Iberoamericana 25 (2009), no. 2, 709--738. https://projecteuclid.org/euclid.rmi/1255440072.


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