Harmonic and quasi-harmonic spheres, Part II
Fang‐Hua Lin, Chang-You Wang · Communications in Analysis and Geometry · 2002
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 [LW2], we will study the rectifiablity of defect measures in the parabolic case as well as the quasi-harmonic sphere bubblings and the so-called generalized varifold flow.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 fields to be bounded in L 2 , that is not a conformally invariant condition.We find an essential optimal condition on tension fields, which is also scaling(up) invariant, and which is always satisfied whenever the tension fields are bounded in LP, 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 9M), N C R k be a compact Riemannian manifold without boundary.For e > 0, let u e G C 2 (M,R k ) be a critical point of the generalized Ginzburg-Landau functional ArU 11