Harmonic and quasi-harmonic spheres. II.
type: Item
instance of: scholarly article
title: Harmonic and quasi-harmonic spheres. II.
zbMATH Open document ID: 1042.58005
author: Changyou Wang
author: Fanghua Lin
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: This paper is in the sequel of the previous work [Commun. Anal. Geom. 7, 397--429 (1999; Zbl 0934.58018)] on the study of approximated harmonic maps in high dimensions. The main purpose of this paper is to understand the bubbling phenomena as well as the energy quantization beyond the natural conformal dimension two for the Dirichelet integral. The authors examine here the static situation, i.e., the studies of harmonic spheres. Let \(M\) be an \(m\)-dimensional compact Riemannian manifold (with possibly non-empty boundary \(\partial M\)), \(N\subset \mathbb R^k\) be a compact Riemannian manifold without boundary. For \(\varepsilon >0\), let \(u_{\varepsilon}\in C^2(M,\mathbb R^k)\) be a critical point of the generalized Ginzburg-Landau functional \(I_{\varepsilon}(u)=\int_M (\frac 12| Du| ^2+\frac 1{\varepsilon ^2}F(u))\,dx,\) where \(F\in C^{\infty}(\mathbb R^k,\mathbb R)\) satisfies that \(F(p)=d^2(p,N),{\text{ if }}d(p,N)\leq \delta;{\text{ or }}=4\delta^2,{\text{ if }}d(p,N)\geq 2\delta .\) Here \(d\) denotes the Euclidean distance in \(\mathbb R^k\) and \(d(\cdot ,N)={\text{inf}}\{d(\cdot ,p):p\in N\}\). Note that \(\delta >0\) is chosen to be so small that \(d^2(\cdot ,N)\) is smooth for \(p\in N_{2\delta}\equiv \{p:d(p,N)\leq 2\delta \}\). It is interesting to study the limit behavior of \(u_{\varepsilon}\)'s, as \(\varepsilon\) tends to zero. In this paper, the authors extend the blow-up techniques developed in [\textit{F. H. Lin}, Ann. Math. (2) 149, 785--829 (1999; Zbl 0949.58017)] to the case that \(N\) does support harmonic \(S^2\). They obtain the bubbling result in the two dimension case, \(m=2\). For \(m\geq 3\), they prove a quantization result for the density function of the defect measure on the concentration set associated with the process of convergence, provided that \(N=S^{k-1}\). These ideas for the generalized Ginzburg-Landau functionals, which are motivated by an earlier work of Helein and some recent works by F. H. Lin and T. Riviere in hi…
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 1893849
zbMATH Keywords: approximated harmonic map
zbMATH Keywords: bounded \(L^2\) tension field
zbMATH Keywords: bubble tree convergence
MaRDI profile type: Publication
DOI: 10.4310/CAG.2002.V10.N2.A5
description: scientific article; zbMATH DE number 1893849