Energy quantization for harmonic maps
type: Item
instance of: scholarly article
title: Energy quantization for harmonic maps
zbMATH Open document ID: 1014.58008
DOI: 10.1215/S0012-7094-02-11116-8
author: Tristan Rivière
author: Fanghua Lin
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: This article presents a higher dimensional, i.e. domain of dimension at least three, energy bubbling theorem: Let \(M\) and \(N\) be Riemannian manifolds, \(\dim{M}\geq 3\). If a sequence \(\{u_{i}\}_{i\in {\mathbb N}}\) of stationary harmonic maps (rather than energy minimising maps) converges towards \(u\), weakly in \(H^{1,1}\), then the \(|\nabla u_{i}|^{2}dx\) converge, as Radon measures, towards \(|\nabla u|^{2}dx + \mu\), where \(u\) is a weakly harmonic map from \(M\) into \(N\), away from a closed rectifiable set \(\Sigma\) of non-zero and finite \((n-2)\)-Hausdorff measure, and \(\mu\) is the contraction with \(\Sigma\) of an \((n-1)\)-dimensional density \(\theta^{n-1}(\mu,x)\) times the \((n-2)\)-Hausdorff measure. The main result of this work is the characterisation (\({\mathcal H}^{n-2}\) a.e.) of \(\theta^{n-1}\) as the sum of energies of smooth harmonic maps from \({\mathbb S}^2\) to \(N\), at least when \(N\) is the standard \(k\)-dimensional sphere, though the authors indicate that this particular choice of target was only made for the sake of clarity. This leads to a corollary on the quantisation of the normalised energy of a stationary harmonic map \(u\) from \({\mathbb R}^3\) into \({\mathbb S}^2\), since then: \[ \lim_{r \to \infty} \frac{1}{r} \int_{B(0,r)} |\nabla u|^{2} dx = 8\pi n \quad (n\in \mathbb N) , \] and if, moreover, \(u\) is smooth: \[ \lim_{r \to \infty} \frac{1}{r} \int_{B(0,r)} |\nabla u|^{2} dx = 16\pi n \quad (n\in \mathbb N). \]
reviewed by: Eric Loubeau
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 1820870
zbMATH Keywords: harmonic maps
zbMATH Keywords: energy quantization
MaRDI profile type: Publication
cites work: On the singular set of stationary harmonic maps
cites work: Q3972645
cites work: Quantization effects for \(-\Delta u=u(1-| u|^ 2)\) in \(\mathbb{R}^ 2\)
cites work: Energy identity for a class of approximate harmonic maps from surfaces
cites work: Minimization of conformally invariant energies in homotopy classes
cites work: Partial regularity for stationary harmonic maps into spheres
cites work: The Dirichlet energy of mappings with values into the sphere
cites work: Q3998311
cites work: Gradient estimates and blow-up analysis for stationary harmonic maps
cites work: Mapping problems, fundamental groups and defect measures
cites work: A quantization property for static Ginzburg-Landau vortices.
cites work: Energy identity of harmonic map flows from surfaces at finite singular time
cites work: Bubbling phenomena of Palais-Smale-like sequences of \(m\)-harmonic type systems
cites work: Bubble tree convergence for harmonic maps
cites work: Pseudo-holomorphic maps and bubble trees
cites work: On singularities of the heat flow for harmonic maps from surfaces into spheres
cites work: The existence of minimal immersions of 2-spheres
cites work: Q3344873
cites work: Q4219625
cites work: Q5690133
cites work: Gromov's Compactness Theorem for Pseudo Holomorphic Curves
full work available at URL: s0012-7094-02-11116-8
OpenAlex ID: W2009322730
description: scientific article; zbMATH DE number 1820870