Harmonic and Quasi-Harmonic Spheres, Part II

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 recti ablity of defect measures in the parabolic case as well as the quasi-harmonic sphere bubblings and the so-called generalized varifold ow.