Blowup behavior of harmonic maps with finite index

TL;DR

This paper investigates the blow-up behavior of sequences of alpha-harmonic maps from Riemann surfaces to manifolds, showing that under certain curvature and bounded index conditions, the energy concentrates along finite geodesics, confirming an energy identity.

Contribution

It establishes conditions under which blow-up limits of alpha-harmonic maps have finite-length geodesic necks and confirms the energy identity in these cases.

Findings

Blow-up limits form finite length geodesics.

Energy identity holds under bounded index and positive Ricci curvature.

Results extend to diverging conformal classes of harmonic maps.

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.