Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphere $\mathbb{S}^3$correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are $4\pi k$, where $k\in\mathbb{N}^*$, with $k\ne 2,3,5,7$. When the ambient space is the four-sphere $\mathbb{S}^4$, the regular homotopy class of immersions of the two-sphere $\mathbb{S}^2$ is determined by the self-intersection number $q\in\mathbb{Z}$; here we shall prove that the possible critical values are $4\pi (\vert q\vert+k+1)$, where $k\in\mathbb{N}$. Moreover, if $k=0$, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration $\mathbb{P}^3\rightarrow \mathbb{S}^4$, from a rational curve in $\mathbb{P}^3$and, if $k\ne 0$, via stereographic projection, from a minimal surface in $\mathbb{R}^4$ with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some $\mathbb{P}^2\subset\mathbb{P}^3$ or (equivalently) when the minimal surface of $\mathbb{R}^4$ is complex with respect to a suitable complex structure of $\mathbb{R}^4$.