Linear growth harmonic functions on complete manifolds
type: Item
instance of: scholarly article
title: Linear growth harmonic functions on complete manifolds
zbMATH Open document ID: 0853.58039
author: Jiaping Wang
published in: Communications in Analysis and Geometry
publication date: 1996-07-03T00:00:00Z
review text: The main task of the author in the present paper is to prove the following theorem: `Let \(M\) be an \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature outside the ball \(B_p(a)\), where \(p \in M\) is a fixed point. Assume that the first Betti number of \(M\) is finite. Then the space of all linear growth harmonic functions on \(M\) has finite dimension. Moreover, if \(\text{Ric}_M \geq - K\) on \(B_p(a)\) for some nonnegative number \(K\), then there exists a constant \(C(K,a,n)\) depending only on \(K\), \(a\) and \(n\) such that the dimension is less than or equal to \(C(K,a,n)\). This result can be considered as a support to the Yau's conjecture on the dimension of the space of harmonic functions on a manifold \(M\) with nonnegative Ricci curvature.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 870994
zbMATH Keywords: complete Riemannian manifold
zbMATH Keywords: harmonic functions
zbMATH Keywords: Ricci curvature
reviewed by: Enrique Outerelo
MaRDI profile type: Publication
Wikidata QID: Q125658978
full work available at URL: cag.1995.v3.n4.a5
OpenAlex ID: W2730747281
DOI: 10.4310/CAG.1995.V3.N4.A5
description: scientific article; zbMATH DE number 870994