Energy identity and removable singularities of maps from a Riemann surface with tension field unbounded in $L^2$

Abstract

We prove the removal singularity results for maps with bounded energy from the unit disk $B$ of $R^2$ centered at the origin to a closed Riemannian manifold whose tension field is unbounded in $L^2(B)$ but satisfies the following condition: {eqnarray*} (\int_{B_t\setminus B_{\frac{t}{2}}}|\tau(u)|^2)^1/2\leq C_1(\frac{1}{t})^a, {eqnarray*} for some $0<a<1$ and for any $0<t<1$, where $C_1$ is a constant independent of $t$. We will also prove that if a sequence $\{u_n\}$ has uniformly bounded energy and satisfies {eqnarray*} (\int_{B_t\setminus B_{\frac{t}{2}}}|\tau(u_n)|^2)^1/2\leq C_2(\frac{1}{t})^a, {eqnarray*} for some $0<a<1$ and for any $0<t<1$, where $C_2$ is a constant independent of $n$ and $t$, then the energy identity holds for this sequence and there will be no neck formation during the blow up process.