Energy Identity for Stationary Harmonic Maps

TL;DR

This paper proves an energy identity for sequences of stationary harmonic maps, showing that the defect measure's energy density equals the sum of energies of bubble maps formed by blow-up analysis.

Contribution

It establishes the energy identity for stationary harmonic maps, linking defect measure energy density to bubble map energies at singular points.

Findings

Energy density equals sum of bubble map energies at singular points.

Defect measure decomposes into bubble maps capturing energy concentration.

Provides a rigorous proof of the energy identity in harmonic map theory.

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 $E[u_j]\equiv \int |\nabla u_j|^2\leq \Lambda$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+\nu$, where $\nu = e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(\nu)$ 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 $\nu$ is precisely the sum of the energies of the bubbling maps.