Are all Dirac-harmonic maps uncoupled?

Abstract

Dirac-harmonic maps $(f,\varphi )$ consist of a map $f:M\to N$ and a twisted spinor $\varphi \in \Gamma (\Sigma M\otimes f^{\ast }TN)$ and they are defined as critical points of the super-symmetric energy functional. A Dirac-harmonic map is called \emph{uncoupled}, if $f$ is a harmonic map. We show that under some minimality assumption Dirac-harmonic maps defined on a closed domain are uncoupled.

see also https://www.berndammann.de/publications/diracharm3 . In version2, some notation was adapted to the standards, which also implied to change the title. In version 1, a boundary term was neglected in Theorem 5 in Section 3. This Section 3 would have needed several modifications and thus was removed in version 2