Energy identity for the maps from a surface with tension field bounded in $L^p$

Abstract

Let $M$ be a closed Riemannian surface and $u_n$ a sequence of maps from $M$ to Riemannian manifold $N$ satisfying $$\underset{n}{\sup }(\Vert \nabla u_n{\Vert }_{L^2(M)}+\Vert \tau (u_n){\Vert }_{L^p(M)})\le \Lambda$$ for some $p>1$, where $\tau (u_n)$ is the tension field of the mapping $u_n$. For the general target manifold $N$, if $p\ge \frac{6}{5}$, we prove the energy identity and neckless during blowing up.