Rigidity of $\varepsilon$-harmonic maps of low degree

In 1981, Sacks and Uhlenbeck introduced their famous $\alpha$-energy as a way to approximate the Dirichlet energy and produce harmonic maps from surfaces into Riemannian manifolds. However, the second and third authors together with Malchiodi ([11],[12]) showed that for maps between two-spheres this method does not capture every harmonic map. They established a gap theorem for $\alpha$-harmonic maps of degree zero and also showed that below a certain energy bound $\alpha$-harmonic maps of degree one are rotations. We establish similar results for $\varepsilon$-harmonic maps $u_\varepsilon \colon S^2\rightarrow S^2$, which are critical points of the $\varepsilon$-energy introduced by the second author in [9]. In particular, we similarly show that $\varepsilon$-harmonic maps of degree zero with energy below $8\pi$ are constant and that maps of degree $\pm 1$ with energy below $12\pi$ are of the form $Rx$ with $R\in O(3)$. Moreover, we construct non-trivial $\varepsilon$-harmonic maps of degree zero with energy $> 8\pi$.