The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature
type: Item
instance of: scholarly article
title: The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature
zbMATH Open document ID: 1005.53027
DOI: 10.1007/BF02773166
author: R. Smith
published in: Israel Journal of Mathematics
publication date: 2003-02-17T00:00:00Z
review text: Let \(\gamma(t)\), \(0\leq t\leq L\) be a minimal geodesic in the Riemannian manifold \(M^n\) with almost non-negative Ricci curvature \(Ric \geq - (n-1)H\), and \(p\) a point such that \(\text{dist}(p,\gamma(t))< \delta L\) for some \(L/3 < t < 2L/3\). In the ball \(B_L(p)\) define a harmonic function \(b\) whose boundary values equal \(\text{dist}(x,\gamma(0))\) for \(x\in \partial B_L(p)\). In the paper under review the author proves that the gradient \(\nabla b\) of the function \(b\) does not vanish if \(L^{-1}, H, \delta\) are sufficiently small and the ball \(B_L(p)\) is close enough in the Gromov-Hausdorff distance to the Euclidean ball of the same radius.
reviewed by: Valery Borisovich Marenich
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 1760718
zbMATH Keywords: Ricci curvature
zbMATH Keywords: harmonic function
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773166
OpenAlex ID: W1967808854
description: scientific article; zbMATH DE number 1760718