Coupled Dirac-harmonic maps from closed spin tori

Abstract

We construct Dirac-harmonic maps from closed spin tori whose map components are not harmonic. For a closed Riemannian surface as target, we prove that such solutions exist for every source metric and every spin structure whenever the target metric is nonflat. The maps can be chosen null-homotopic, of rank one everywhere, with nowhere-vanishing tension and unbounded energy. For a strictly negatively curved closed oriented target surface, every homotopy class contains such a sequence when the source has the trivial spin structure. For the other spin structures we give a compatibility condition for the same construction. Adding a harmonic circle component yields coupled immersions into the product of any such surface with a circle. The construction combines a spinor phase that cancels the rotation of a moving target frame with a curvature balance along a closed curve. Minimizing the energy of short loops with prescribed signed curvature area produces the required curves without any symmetry assumption on the target. Their total geodesic curvature determines the spinor's periodicity. We also give a curve criterion in arbitrary target dimension, applications to totally geodesic surfaces and warped products, and a local choice of target metric that works on every closed manifold of dimension at least two.