Liouville Theorems for Dirac-Harmonic Maps

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space ${\mathbb{R}}^n$, the hyperbolic space \H^n and a Riemannian manifold ${\mathfrak{S}}^{\mathfrak{n}}$ ($n\ge 3$) with the Schwarzschild metric to any Riemannian manifold $N$.