Energy Identity for Stationary Harmonic Maps

Abstract

In this paper, we consider sequences $u_j:B_2\subseteq M\rightarrow N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\le \Lambda $. After passing to a subsequence, it is known one can limit $u_j\rightarrow u:B_1\rightarrow N$ with the associated defect measure $|\nabla u_j|^2 dv_g \rightarrow |\nabla u|^2dv_g+\nu $, where $\nu = e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure (Lin in Ann. Math. (2) 149(3):785–829, 1999). For a.e. $x\in S=\operatorname {supp}(\nu )$, one can produce a finite number of bubble maps $b_j:S^2\rightarrow N$ 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. The analysis requires several new ideas. The energy near a blow up region can be naturally decomposed $|\nabla u|^2 = |\pi _L u|^2+ |\pi _{L^\perp } u|^2= |\pi _L u|^2+ \langle \nabla u, n^\perp \rangle ^2+ \langle \nabla u, \alpha \rangle ^2$ into the $L^{m-2}$-symmetric and $L^\perp $-bubble directions, with the 2-dimensional $L^\perp $-bubble directions themselves broken down into radial and angular components. Each of the three components of the energy requires a new point. In the symmetric L-energy directions, we are able to provide a new log-improved energy bound, and for the angular $\alpha $-energy, we produce a new superconvexity estimate. The most delicate new piece of analysis is needed for the radial n-energy component, where a priori the errors for the energy identity strictly larger than what is allowed. To handle this, we will approximate the $m-2$ blow up set by a submanifold $\mathcal T$ which solves an appropriate Euler–Lagrange equation. This can be done so that these highest order errors are canceled by the equation for $\mathcal T$, at least up to controllable terms.

Keywords

Quantitative Stratification / Neck regions / Annular regions / Energy identity / 35J60 / 53C43 / 58E20

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