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

## TL;DR

This paper extends the concept of $eta$-harmonic maps to $eta$-Dirac-harmonic maps, establishing existence, regularity, and convergence results for these maps from closed surfaces into manifolds with nonpositive curvature.

## Contribution

It introduces $eta$-Dirac-harmonic maps, proves their existence under certain curvature conditions, and demonstrates their smoothness and convergence properties.

## Key findings

- Existence of nontrivial perturbed $eta$-Dirac-harmonic maps for nonpositive curvature targets.
- Proved regularity and smoothness of these maps.
- Established convergence of perturbed maps to smooth $eta$-Dirac-harmonic maps.

## 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 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.