A weak energy identity and the length of necks for a sequence of Sacks-Uhlenbeck α-harmonic maps

In this paper we discuss the convergence behavior of a sequence of alpha-harmonic maps u(alpha) with E(alpha)(u(alpha)) < C from a compact surface (M, g) into a compact Riemannian manifold (N, h) without boundary. Generally, such a sequence converges weakly to a harmonic map in W(1,2)(M, N) and strongly in C(infinity) away from a finite set of points in M. If energy concentration phenomena appears, we show a generalized energy identity and discover a direct convergence relation between the blow-up radius and the parameter alpha which ensures the energy identity and no-neck property. We show that the necks converge to some geodesics. Moreover, in the case there is only one bubble, a length formula for the neck is given. In addition, we also give an example which shows that the necks contain at least a geodesic of infinite length. (C) 2010 Elsevier Inc. All rights reserved.