Energy Quantization for Harmonic Maps

In this paper we establish the higher-dimensional energy bubbling results for harmonic maps to spheres. We have shown in particular that the energy density of concentrations has to be the sum of energies of harmonic maps from the standard 2dimensional spheres. The result also applies to the structure of tangent maps of stationary harmonic maps at either a singularity or infinity. 0. Introduction Let M , N be smooth, compact Riemannian manifolds without boundary. Suppose u : M → N is a smooth harmonic map such that the homotopy class, [u], of u is not trivial. Then it follows easily from the small energy regularity theorem of R. Schoen and K. Uhlenbeck (cf. [Sc]) that the total energy of the map u is