Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the \(\alpha _k\)-harmonic map sequences with bounded uniformly \(\alpha _k\)-energy, denoted by \(\{u_{\alpha _k}: \alpha _k>1 \quad \text{ and } \quad \alpha _k\searrow 1\}\), 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 \(\alpha _k\searrow 1\), 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.