  In this paper, we study the blow-up phenomena on the $\alpha_k$-harmonic map
sequences with bounded uniformly $\alpha_k$-energy, denoted by $\{u_{\alpha_k}:
\alpha_k>1 \quad \mbox{and} \quad \alpha_k\searrow 1\}$, from a compact Riemann
surface into a compact Riemannian manifold. If the Ricci curvature of the
target manifold is of 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.
