Energy identity for Ginzburg-Landau approximation of harmonic maps
Xuanyu Li
TL;DR
The paper proves a full energy identity for the Ginzburg–Landau approximation of harmonic maps with a general target, extending known results beyond two dimensions or spheres by adapting the Naber–Valtorta framework. It develops a refined bubble–annulus decomposition around the defect set, introduces heat-mollified energy, and constructs local best planes and best-approximating submanifolds to control tangential, angular, and radial energy in annular regions. By establishing angular energy convexity and vanishing energy in annuli through cone-splitting and a gluing construction, it shows the defect measure $\nu$ decomposes into a sum of harmonic-sphere energies on the bubble tree, i.e., $\nu=\sum_j \frac{1}{2}\int_{\mathbb{S}^2}|\nabla u_j|^2$. This work thereby provides a precise local model for energy concentration in high dimensions and strengthens the connection between Ginzburg–Landau approximations and the fine structure of stationary 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 $$E_ε(u)=\int_M\frac{\vert\nabla u\vert^2}{2}+\frac{F(u)}{ε^2},u\in W^{1,2}(M,\mathbb{R}^L),$$ 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 $ε\rightarrow0$, the critical points $u_ε$ of $E_ε$ with energy bound $E_ε(u_ε)\leqslantΛ$ 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 $$\left(\frac{\vert\nabla u_ε\vert^2}{2}+\frac{F(u_ε)}{ε^2}\right)dx\rightarrow\frac{\vert\nabla u_ε\vert^2}{2}dx+ν,$$ and the defect measure $ν$ above is $(dimM-2)$-rectifiable. Lin-Wang showed that if $N$ is a sphere or dim$M$=2, then the density of $ν$ 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.