Energy Identity for Stationary Harmonic Maps

Abstract

In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j{|}^2\le \hat{I}›$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j{|}^{2}d{v}_g\to |\nabla u{|}^{2}d{v}_g+\hat{I}½$, where $\hat{I}½=e(x)H_S^{m-2}$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\mathrm{supp}(\hat{I}½)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)={\sum }_{j}E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $\hat{I}½$ is precisely the sum of the energies of the bubbling maps.