Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the alpha k-harmonicmap sequences with bounded uniformly alpha k-energy, denoted by {u(alpha k) : alpha k > 1 and alpha k SE arrow 1}, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the alpha k-harmonic map sequence with respect to the corresponding alpha k-energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of {u(alpha k)} as alpha k SE arrow 1, the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence u(k) : (Sigma, h(k)) -> N, where the conformal class defined by h(k) diverges, we also prove some similar results.