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Removable singularities of harmonic maps. (English) Zbl 0559.58011
Let M,N be Riemannian manifolds. The removable singularity problem is as follows: Let \(f\in L^ 2_{1,loc}(M,N)\) be weakly harmonic on \(M\setminus A\), where \(A\subset M\) is relatively closed. How large is the set A such that f is harmonic on M, if f is given in a certain class of maps? The authors consider the classes: \(L^ p_{1,loc}(M,N)\), \(C^ 0(M,N)\), \(C^{\alpha}(M,N)\), \(C^ 1(M,N)\), \(C^{1,\alpha}(M,N)\), \(0<\alpha \leq 1\), etc. for f, and apply the concepts of \(C_ q\)-capacity due to Choquet, and the Hausdorff measure to measure the size of the set A. Certain necessary and sufficient conditions are obtained. These results are independent of the Riemannian metric on M. A similar result for \(L^ p_{1,loc}\) class and nonpositively sectional curved N has been obtained by M. Meier [J. Reine Angew. Math. 344, 87-101 (1983; Zbl 0521.35018)]. In addition, a version of the Schwarz reflection principle for harmonic maps, generalizing that given by J. C. Wood, is also given.
Reviewer: K.Chang

MSC:
58E20 Harmonic maps, etc.
58C35 Integration on manifolds; measures on manifolds
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