No neck for approximate harmonic maps to the sphere

In this paper, we consider a sequence of maps from a Riemann surface to a standard sphere with tension fields bounded in an Orlicz space ϕ(L)ϕ(L) where ϕ(L)={f:∫ϕ(∣f∣)<∞}.ϕ(L)={f:∫ϕ(∣f∣)<∞}. If there holds limt→∞ϕ(t)tlnt=∞, we can prove that there is no neck during blow up. This result improves our previous theorems in Li and Zhu [1] and [2]. One can see that our result here is optimal in the size conditions.