Surfaces in ${\mathbb{S}}^4$ with normal harmonic Gauss maps

We consider conformal immersions of Riemann surfaces in $\text{bb}{S}^4$ and study their Gauss maps with values in the Grassmann bundle $$\mathcal{F}$ = SO_5/T^2 \to $\mathbb{S}$^4$\mathrm{.}TheenergyofmapsfromRiemannsurfacesinto$$\mathcal{F}$$ is considered with respect to the normal metric on the target and immersions with harmonic Gauss maps are characterized. We also show that the normal-harmonic map equation for Gauss maps is a completely integrable system, thus giving a partial answer of a question posed by Y. Ohnita in \cite{ohnita}. Associated ${\mathbb{S}}^1$-families of parallel mean curvature immersions in ${\mathbb{S}}^4$ are considered. A lower bound of the normal energy of Gauss maps is obtained in terms of the genus of the surface.