Let
un be a sequence of mappings from a closed Riemannian surface
M to a general Riemannian manifold
N . If
un satisfies
where
τ(un) is the tension field of
un, then the so called energy identity and neckless property hold during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in
L1(M) but the energy identity fails.