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

TL;DR

This paper establishes removal singularity results and energy identity for maps from a Riemann surface with unbounded tension fields in $L^2$, under specific growth conditions, ensuring no energy loss or neck formation during blow-up.

Contribution

It proves removal of singularities and energy identity for harmonic maps with unbounded tension fields satisfying a specific growth condition, extending classical results.

Findings

Removal of singularities under unbounded tension fields.

Energy identity holds for sequences with bounded energy and growth conditions.

No neck formation occurs during blow-up under the given conditions.

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.