Blowup behavior of harmonic maps with finite index

Abstract

In this paper, we study the blow-up phenomena on the alpha _k-harmonic map sequences with bounded uniformly alpha _k-energy, denoted by {u_{alpha _k}: alpha _k>1 quad text{ and } quad alpha _ksearrow 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 _ksearrow 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)rightarrow N, where the conformal class defined by h_k diverges, we also prove some similar results.