Harmonic maps with defects. (English) Zbl 0608.58016

The authors study the problem of minimizing energy in certain classes C of maps \(v: {\mathbb{R}}^ 3\supset \Omega \to S^ 2\) defined on some 3- dimensional region \(\Omega\) taking values in the two-sphere \(S^ 2\). In the first case C consists of all functions v which are continuous except for a finite number of points \(x_ 1,...,x_{\ell}\) in \(\Omega\) with prescribed degree \(\deg (v,x_ i)=d_ i\) and finite energy \(E(v):=\int_{\Omega}| Dv|^ 2 dx\). Then it is shown that \[ (1)\quad \inf_{v\in C}E(v)=8\pi L, \] where L is a geometric quantity depending on \(\Omega\), the distances between the points \(x_ i\) and on the degrees \(d_ i\). The upper bound follows by constructing an almost minimizer, the proof of the lower bound is more involved and based on the analysis of the so-called D-field associated to \(v\in C\). Moreover, equation (1) is extended to the case that the point singularities \(x_ i\) are replaced by ”singular holes” (i.e. disjoint compact subsets of \(\Omega)\). The second part of the paper deals with the Dirichlet problem for maps \(v: {\mathbb{R}}^ 3\supset \Omega \to S^ 2\) with prescribed boundary function \(g: \partial \Omega \to S^ 2\). By a result of R. Schoen and K. Uhlenbeck [J. Differ. Geom. 17, 307-335, correction ibid. 19, 329 (1982; Zbl 0521.58021); 18, 253-268 (1983; Zbl 0547.58020)] a minimizer u is smooth except for a finite number of singular points \(x_ 1,...,x_ k\). It is shown that \(\deg (u,x_ i)=\pm 1\) and that (near \(x_ i)\) u behaves like \(R(x+x_ i)/| x-x_ i|\) for a rotation R. This result is a consequence of another theorem saying that in case \(\Omega =B^ 3\) the homogeneous extension g(x/\(| x|)\) is E minimizing for boundary values g iff g is constant or a rotation. In a final chapter the authors replace \({\mathbb{R}}^ 3\), \(S^ 2\), \(\int | Dv|^ 2\) by \({\mathbb{R}}^ n\), \(S^{n-1}\) and \(\int | Dv|^{n-1}\) for some \(n\geq 3\) and prove analogous theorems.
Reviewer: M.Fuchs


58E20 Harmonic maps, etc.
49Q99 Manifolds and measure-geometric topics
Full Text: DOI


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