Energy quantization for harmonic maps
This article presents a higher dimensional, i.e. domain of dimension at least three, energy bubbling theorem: Let \(M\) and \(N\) be Riemannian manifolds, \(\dim{M}\geq 3\). If a sequence \(\{u_{i}\}_{i\in {\mathbb N}}\) of stationary harmonic maps (rather than energy minimising maps) converges towards \(u\), weakly in \(H^{1,1}\), then the \(|\nabla u_{i}|^{2}dx\) converge, as Radon measures, towards \(|\nabla u|^{2}dx + \mu\), where \(u\) is a weakly harmonic map from \(M\) into \(N\), away from a closed rectifiable set \(\Sigma\) of non-zero and finite \((n-2)\)-Hausdorff measure, and \(\mu\) is the contraction with \(\Sigma\) of an \((n-1)\)-dimensional density \(\theta^{n-1}(\mu,x)\) times the \((n-2)\)-Hausdorff measure. The main result of this work is the characterisation (\({\mathcal H}^{n-2}\) a.e.) of \(\theta^{n-1}\) as the sum of energies of smooth harmonic maps from \({\mathbb S}^2\) to \(N\), at least when \(N\) is the standard \(k\)-dimensional sphere, though the authors indicate that this particular choice of target was only made for the sake of clarity. This leads to a corollary on the quantisation of the normalised energy of a stationary harmonic map \(u\) from \({\mathbb R}^3\) into \({\mathbb S}^2\), since then: \[ \lim_{r \to \infty} \frac{1}{r} \int_{B(0,r)} |\nabla u|^{2} dx = 8\pi n \quad (n\in \mathbb N) , \] and if, moreover, \(u\) is smooth: \[ \lim_{r \to \infty} \frac{1}{r} \int_{B(0,r)} |\nabla u|^{2} dx = 16\pi n \quad (n\in \mathbb N). \]
scientific article; zbMATH DE number 1820870