On conformal Gauss maps

We characterise the maps into the space of $2$-spheres in $S^n$ that are the conformal Gauss maps of conformal immersions of a surface. In particular, we give an invariant formulation and efficient proof of a characterisation, due to Dorfmeister–Wang \cites{DorWan13,DorWan}, of the harmonic maps that are conformal Gauss maps of Willmore surfaces.