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MiS Preprint

$\alpha$-Dirac-harmonic maps from closed surfaces

Jürgen Jost and Jingyong Zhu


$\alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $\alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $\alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $\alpha$-harmonic maps for $\alpha >1$ and then letting $\alpha \to 1$. 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 $\alpha$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $\varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $\alpha$-Dirac-harmonic maps converges to a smooth nontrivial $\alpha$-Dirac-harmonic map.

Mar 19, 2019
Mar 29, 2019
MSC Codes:
58E05, 58E20
Palais-Smale condition, $\alpha$-Dirac-harmonic map, nonlinear perturbation

Related publications

2021 Journal Open Access
Jürgen Jost and Jingyong Zhu

\(\alpha\)-Dirac-harmonic maps from closed surfaces

In: Calculus of variations and partial differential equations, 60 (2021) 3, p. 111