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 $\{u_{\alpha_k}: \alpha_k>1 \quad \mbox{and} \quad \alpha_k\searrow 1\}$, 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.