Energy Identity for Stationary Harmonic Maps
Aaron Naber, Daniele Valtorta
TL;DR
The paper establishes the energy identity for sequences of stationary harmonic maps with uniformly bounded energy by showing the defect measure decomposes exactly into bubbling energies at almost every bubbling point. It introduces a quantitative energy identity asserting that, on most 2D slices aligned with a prescribed $(m-2)$-dimensional symmetry, the maps are ε-harmonic with a finite bubble decomposition, and the limit accounts for all defect energy. The core strategy combines a quantitative annulus/bubble decomposition with a meticulous analysis of annular regions, using heat-kernel mollified energies, restricted energy functionals, and best-approximating submanifolds to control energy transfer between scales. The approach also develops a robust framework of local and global approximations (best planes and submanifolds) to cancel high-order errors and to establish L^2-energy identities on annular regions, culminating in the energy identity for stationary harmonic maps and its quantitative version with implications for bubbling analysis and rectifiability. The results advance the understanding of energy concentration in codimension-two singular sets and provide tools that mirror Yang–Mills techniques in a harmonic-map setting, adapted to the distinctive two-dimensional Green’s function behavior.
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 $E[u_j]\equiv \int |\nabla u_j|^2\leq Λ$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+ν$, where $ν= e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(ν)$ 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 $ν$ is precisely the sum of the energies of the bubbling maps.