Vol. 14, No. 3, 2021

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A gap theorem for $\alpha$-harmonic maps between two-spheres

Tobias Lamm, Andrea Malchiodi and Mario Micallef

Vol. 14 (2021), No. 3, 881–889
Abstract

We consider approximations introduced by Sacks and Uhlenbeck of the harmonic energy for maps from ${S}^{2}$ into ${S}^{2}$. We continue our previous analysis (J. Differential Geom. 116:2 (2020), 321–348) on limits of $\alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity of Li and Zhu (Ann. Inst. H. Poincaré Anal. Non Linéaire 36:1 (2019), 103–118), we obtain an optimal gap theorem for the $\alpha$-harmonic maps of degree $-1,0$ or $1$.

Keywords
$\alpha$-harmonic maps, gap theorems
Primary: 58E20
Milestones
Received: 25 March 2019
Accepted: 21 November 2019
Published: 18 May 2021
Authors
 Tobias Lamm Institute for Analysis Karlsruhe Institute of Technology Karlsruhe Germany Andrea Malchiodi Scuola Normale Superiore Pisa Italy Mario Micallef Mathematics Institute University of Warwick Coventry United Kingdom