Harmonic and Quasi - Harmonic Spheres , Part

This is in the sequel of our previous work LW] on the study of the approximated harmonic maps in high dimensions. The main purpose of the present article is to understand the bubbling phenomena as well as the energy quantization beyond the natural conformal dimension two for the Dirichelet integral. This will be important toward our understandings of the defect measures and the energy concentration sets introduced and studied already for approximated harmonic maps in LW]. We shall examine here the static situation, that is, the studies of harmonic spheres. In our forthcoming work, we will study the rectiiablity of defect measures in the parabolic case as well as the quasi-harmonic sphere bubblings and the so-called generalized varifold ow. As bi-products of our study are improvements of the \energy identity"as well as the \no necks formations" thorems for approximated harmonic maps from Riemannian surfaces. In all previous works one needs to assume the tension elds to be bounded in L 2 , that is not a conformally invariant condition. We nd an essential optimal condition on tension elds, which is also scaling(up) invariant, and which is always satissed whenever the tension elds are bounded in L p , for any p > 1. To describe the main results more precisely, we let M be a m dimensional compact Riemannian manifold (with possibly non-empty boundary @M), N R k be a compact Riemannian manifold without boundary. For > 0, let u 2 C 2 (M; R k) be a critical point of the generalized Ginzburg-Landau functional