Energy identity for Ginzburg-Landau approximation of harmonic maps

Abstract: Given two Riemannian manifolds $M$ and $N\subset\mathbb{R}^L$, we consider the energy concentration phenomena of the penalized energy functional $$E_{\epsilon}(u)=\int_M\frac{\vert\nabla u\vert^2}{2}+\frac{F(u)}{\epsilon^2},u\in W^{1,2}(M,\mathbb{R}^L),$$ where $F(x)$=dist$(x,N)$ in a small tubular neighborhood of $N$ and is constant away from $N$. It was shown by Chen-Struwe that as $\epsilon\rightarrow0$, the critical points $u_{\epsilon}$ of $E_{\epsilon}$ with energy bound $E_{\epsilon}(u_{\epsilon})\leqslant\Lambda$ subsequentially converge weakly in $W^{1,2}$ to a weak harmonic map $u:M\rightarrow N$ . In addition, we have the convergence of the energy density $$\left(\frac{\vert\nabla u_{\epsilon}\vert^2}{2}+\frac{F(u_{\epsilon})}{\epsilon^2}\right)dx\rightarrow\frac{\vert\nabla u_{\epsilon}\vert^2}{2}dx+\nu,$$ and the defect measure $\nu$ above is $(dimM-2)$-rectifiable. Lin-Wang showed that if $N$ is a sphere or dim$M$=2, then the density of $\nu$ can be expressed by the sum of energies of harmonic spheres. In this paper, we prove this result for an arbitrary $M$ using the idea introduced by Naber-Valtorta.