alpha-Dirac-harmonic maps from closed surfaces

TLDR: In this paper, the existence of nontrivial perturbed Dirac-harmonic maps when the target manifold has non-positive curvature was proved and the regularity theorem showed that they are actually smooth if the perturbation function is smooth.

Abstract: $$\alpha $$ -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 closed surfaces. For $$\alpha >1$$ , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ -harmonic maps for $$\alpha >1$$ and then letting $$\alpha \rightarrow 1$$ . 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 if the perturbation function is 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 coupled $$\alpha $$ -Dirac-harmonic map.