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 $$\sup_n(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^p(M)})\leq \Lambda$$ for some $p>1$, where $\tau(u_n)$ is the tension field of the mapping $u_n$. For the general target manifold $N$, if $p\geq \frac 65$, we prove the energy identity and neckless during blowing up.