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AN OBSERVATION ON THE DIRICHLET PROBLEM AT INFINITY IN RIEMANNIAN CONES

Part of: Other generalizations Global differential geometry

Published online by Cambridge University Press:  22 November 2022

JEAN C. CORTISSOZ Open the ORCID record for JEAN C. CORTISSOZ [Opens in a new window]
JEAN C. CORTISSOZ*
Affiliation:
Department of Mathematics, Universidad de los Andes, Bogotá DC, Colombia
*
jcortiss@uniandes.edu.co
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Abstract

In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.

MSC classification

Primary: 31C05: Harmonic, subharmonic, superharmonic functions
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 250 , June 2023 , pp. 352 - 364
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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