Partial regularity for stationary harmonic maps into spheres

In an interesting recent paper [12], F. HI~LEIN has shown that any weakly harmonic mapping from a two-dimensional surface into a sphere is smooth. I present here a kind of generalization to higher dimensions, asserting in effect that a stationary harmonic mapping from an open subset of Nn(n __> 3) into a sphere is smooth, except possibly for a closed singular set of (n 2 ) dimensional Hausdorff measure zero. My proof crucially depends upon several of H~L~IN's observations (as streamlined by P.-L. LIoNs). To state the result precisely let us suppose that m, n => 2, U is a smooth open subset of Nn, and S m-1 denotes the unit sphere in Nm. A function u in the Sobolev space HI (U;Rm) , u = (u 1 . . . . ,urn), belongs to H I ( U ; S ml ) provided t u[ = 1 a.e. in U

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