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 maps of the form u R (x) =Rx,R∈O(3), are the only critical points of E α for maps from S2 to S2 whose α-energy lies below some threshold. In particular, nontrivial dilations (which are harmonic) cannot arise as strong limits of α-harmonic maps.

中文翻译：

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$\alpha$-谐波映射的限制

众所周知，像 Sacks-Uhlenbeck 一样的 Dirichlet 能量近似的临界点在适当的意义上会收敛到谐波映射。然而，我们表明并不是每个谐波映射都可以通过这种扰动能量的临界点来近似。事实上，我们证明了常数映射和形式为 u R (x) =Rx,R∈O(3) 的映射是 E α 的唯一临界点，用于从 S2 到 S2 的映射，其 α 能量低于某个阈值。特别是，非平凡膨胀（谐波）不能作为 α-谐波映射的强限制出现。