Dirac-geodesics and their heat flows

Dirac-geodesics are Dirac-harmonic maps from one dimensional domains. In this paper, we introduce the heat flow for Dirac-geodesics and establish its long-time existence and an asymptotic property of the global solution. We classify Dirac-geodesics on the standard 2-sphere \(S^2(1)\) and the hyperbolic plane \(\mathbb {H}^2\), and derive existence results on topological spheres and hyperbolic surfaces. These solutions constitute new examples of coupled Dirac-harmonic maps (in the sense that the map part is not simply a harmonic map).