Harmonic and quasi-harmonic spheres. II.
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 higher dimensions, are used by authors to extend the known results on the energy identity and the bubble tree convergence for approximated harmonic maps from surfaces with bounded \(L^2\) tension field to the case that the tension field of the approximated harmonic maps from surfaces with bounded \(L^p\) tension field for any \(p>1\), provided that \(N=S^{k-1}\).
scientific article; zbMATH DE number 1893849