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A gap theorem for \(\alpha\)-harmonic maps between two-spheres. (English) Zbl 1477.58008

For a real number \(\alpha>1\), \(\alpha\)-harmonic maps are critical points of the functional \[ E_\alpha(u) = \frac12 \int_M (2+|\nabla u|^2)^\alpha dA_M, \] where \(u \in W^{1, 2\alpha}(M, N)\) and \(M\) is a compact Riemann surface without boundary and \(N\) is a compact Riemannian manifold. The paper under review which is handling a gap property for \(\alpha\)-harmonic maps is the continuation of their paper [the first author et al., J. Differ. Geom. 116, No. 2, 321–348 (2020; Zbl 1509.53074)]. Assume both \(M\) and \(N\) are \(S^2\), the standard \(2\)-sphere. In their previous paper, the authors proved that for fixed \(\eta>0\), there exists \(\bar \alpha - 1>0\) small, \(\bar \alpha\) depending only on \(\eta\), such that if \(1 < \alpha \le \bar \alpha\) and \(u : S^2 \to S^2\) is \(\alpha\)-harmonic, of degree zero and \(E_\alpha(u) \le 8\pi - \eta\), then \(u\) is constant. This result is an improved version of [J. Sacks and K. Uhlenbeck, Ann. Math. (2) 113, 1–24 (1981; Zbl 0462.58014)].
In the present paper, the authors improve Theorem 1.3 in [the first author et al., loc. cit.] quantitatively by exploiting some bubbling estimates of new type from [J. Li and X. Zhu, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 36, No. 1, 103–118 (2019; Zbl 1410.58004)]. More precisely, they prove that for fixed \(\eta>0\), there exists \(\bar \alpha - 1>0\) so that the only critical points \(u_\alpha\) of \(E_\alpha\) of degree \(\pm 1\) which satisfy \(E_\alpha(u_\alpha) \le 8^\alpha 2\pi - \eta\) and \(1 < \alpha \le \bar \alpha\) are maps of the form \(u^R(x) = Rx\), with \(R\in \mathrm{O}(3)\).
Finally, they show that for every \(\epsilon>0\), there exists \(\alpha_0>1\) which depends only on \(\epsilon\) such that if \(1< \alpha <\alpha_0\) there exists an \(\alpha\)-harmonic map \(u_\alpha:S^2 \to S^2\) with \(\mathrm{deg}(u_\alpha) = 0\) and \(6^\alpha 2\pi \le E_\alpha(u_\alpha) < 6^\alpha 2\pi + \epsilon\). This result implies that the previous gap property on the \(\alpha\)-harmonic map of degree zero is optimal in some sense.

MSC:

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
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References:

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