Remarks on approximate harmonic maps in dimension two

Abstract

For the class of approximate harmonic maps \(u\in W^{1,2}(\Sigma ,N)\) from a closed Riemmanian surface \((\Sigma ,g)\) to a compact Riemannian manifold (N, h), we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps \(\{u_n\}:\Sigma \rightarrow N\), with tension fields \(\tau (u_n)\) bounded in the Morrey space \(M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\); and (ii) if an approximate harmonic map u has tension field \(\tau (u)\in L\log L(\Sigma )\cap M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\), then \(u\in W^{2,1}(\Sigma , N)\). Based on these estimates, we further establish the bubble tree convergence, referring to energy identity both \(L^{2,1}\) of gradients and \(L^1\)-norm of hessians and the oscillation convergence, for a weakly convergent sequence of approximate harmonic maps \(\{u_n\}\), with tension fields \(\tau (u_n)\) uniformly bounded in \(M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\) and uniformly integrable in \(L\log L(\Sigma )\).