Rank zero and rank one sets of harmonic maps
type: Item
instance of: scholarly article
title: Rank zero and rank one sets of harmonic maps
zbMATH Open document ID: 1006.58011
author: Qing Han
author: Fanghua Lin
published in: Methods and Applications of Analysis
publication date: 2002-06-19T00:00:00Z
review text: This paper studies the sizes of the sets of points where a harmonic map from a ball in \(\mathbb{R}^n\) to a smooth compact Riemannian manifold \(N\) is of rank zero or of rank one. This generalizes the question of nonsingular nodal sets and critical nodal sets which have been studied intensively for harmonic functions, and, more generally, for solutions of elliptic equations of the second order. The results are as follows. If \(u\) is either a smooth or a minimizing harmonic map from the ball to \(N\), then the rank zero set \(R_0(u)\) is countably \((n{-}2)\)-rectifiable unless \(u\) maps the whole ball to a point in \(N\). Also, the rank one set \(R_1(u)\) is countably \((n{-}1)\)-rectifiable unless \(u\) maps the whole ball to a point or a geodesic in \(N\). There are bounds for the corresponding Hausdorff measures of \(R_0(u)\) and \(R_1(u)\), at least if intersected with the ball of half size, which depend on the manifold \(N\), the supnorm of the gradient of \(u\) (in the smooth case) or the total energy of \(u\), and some term which controls the oscillation of \(u\) (in the rank zero case) or the deviation of \(u\) from geodesic maps (in the rank one case).
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 1757309
zbMATH Keywords: minimizing harmonic maps
zbMATH Keywords: rectifiability
zbMATH Keywords: nodal sets
MaRDI profile type: Publication
full work available at URL: maa.2000.v7.n2.a9
OpenAlex ID: W134811010
DOI: 10.4310/MAA.2000.V7.N2.A9
description: scientific article; zbMATH DE number 1757309