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

TL;DR

This paper investigates the rigidity and classification of low-degree $ ext{epsilon}$-harmonic maps from the 2-sphere to itself, establishing energy thresholds below which maps are trivial or rotations, and constructing non-trivial examples above these thresholds.

Contribution

The authors extend gap theorems for $ ext{epsilon}$-harmonic maps of low degree, providing energy bounds for triviality and rotational symmetry, and construct non-trivial maps exceeding these bounds.

Findings

$ ext{epsilon}$-harmonic maps of degree zero with energy below $8\pi$ are constant.

Maps of degree $ ext{±}1$ with energy below $12\pi$ are rotations.

Existence of non-trivial degree zero $ ext{epsilon}$-harmonic maps with energy greater than $8\pi$.

Abstract

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 $> 8\pi$.