Limits of $\alpha$-harmonic maps

TL;DR

This paper investigates the limitations of approximating harmonic maps via $\alpha$-harmonic maps, showing that only trivial solutions like constant maps and rotations are attainable as limits under certain energy constraints.

Contribution

It demonstrates that not all harmonic maps can be approximated by critical points of the perturbed energies, identifying specific maps that are the only critical points below a certain energy threshold.

Findings

Constant maps and rotations of $S^2$ are the only critical points below the energy threshold.

Nontrivial dilations cannot be obtained as strong limits of $\alpha$-harmonic maps.

Certain harmonic maps are unreachable via the $\alpha$-energy approximation.

Abstract

Critical points of approximations of the Dirichlet energy \`{a} la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are the only critical points of $E_{\alpha}$ for maps from $S^2$ to $S^2$ whose $\alpha$-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of $\alpha$-harmonic maps.