α-Dirac-harmonic maps from closed surfaces
Jürgen Jost and Jingyong Zhu
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Submission date: 19. Mar. 2019
MSC-Numbers: 58E05, 58E20
Keywords and phrases: Palais-Smale condition, $\alpha$-Dirac-harmonic map, nonlinear perturbation
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α-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to α-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For α > 1, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing α-harmonic maps for α > 1 and then letting α → 1. The extension of this scheme to Dirac-harmonic maps meets with several diﬃculties, and in this paper, we start attacking those. We ﬁrst prove the existence of nontrivial perturbed α-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By 𝜀-regularity and suitable perturbations, we can then show that such a sequence of perturbed α-Dirac-harmonic maps converges to a smooth nontrivial α-Dirac-harmonic map.