A gap theorem for α-harmonic maps between two-spheres

Lamm, Tobias; Malchiodi, Andrea; Micallef, Mario

doi:10.2140/apde.2021.14.881

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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

DOI: 10.2140/apde.2021.14.881

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 $α$ -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 $α$ -harmonic maps of degree $- 1, 0$ or $1$ .

Keywords

$\alpha$-harmonic maps, gap theorems

Mathematical Subject Classification 2010

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

Vol. 14, No. 3, 2021