NONDEGENERACY OF HARMONIC MAPS FROM R-2 TO S-2

We prove that all harmonic maps from R-2 to S-2 with finite energy are nondegenerate. That is, for any harmonic map u from R-2 to S-2 of degree m is an element of Z, all bounded kernel maps of the linearized operator L-u at u are generated by those harmonic maps near u and hence the real dimension of bounded kernel space of L-u is 4 vertical bar m vertical bar + 2.