Harmonic functions of polynomial growth on complete manifolds. II
type: Item
instance of: scholarly article
title: Harmonic functions of polynomial growth on complete manifolds. II
zbMATH Open document ID: 0843.53033
author: Atsushi Kasue
published in: Journal of the Mathematical Society of Japan
publication date: 1996-08-15T00:00:00Z
review text: A complete noncompact Riemannian manifold \(M\) of dimension \(m\) has the strong Liouville property if the dimension of the space \({\mathcal H}_d (M)\) of harmonic functions of polynomial growth of order \(\leq d\) is finite for all \(d\). The author shows that if \(M\) is a Hadamard manifold with volume growth \(V(B (x_0, r))\leq \text{const}\cdot r^m\) and with sectional curvature \(K\geq- \text{const}/ r^2\), then \(M\) has the strong Liouville property. He also shows a corresponding result in the case of nonnegative curvature. In the latter case, stronger results have been announced by T. Colding and W. Minicozzi.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 844441
zbMATH Keywords: Liouville property
zbMATH Keywords: harmonic functions
zbMATH Keywords: Hadamard manifold
zbMATH Keywords: volume growth
zbMATH Keywords: sectional curvature
reviewed by: Werner Ballmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04710037
OpenAlex ID: W1987483150
DOI: 10.2969/JMSJ/04710037
description: scientific article; zbMATH DE number 844441