Gauss maps of M\"obius surfaces in the $n$-dimensional sphere

In this note we discuss Gauss maps for M\"obius surfaces in the $n$-sphere, and their applications in the study of Willmore surfaces. One such ``Gauss map'', naturally associated to a Willmore surface that has a dual Willmore surface, is the Lorentzian $2$-plane bundle given by a lift of the suface and its dual. More generally, we define the concept of a Lorentzian $2$-plane lift for an arbitrary M\"obius surface, and show that the conformal harmonicity of this lift is equivalent to the Willmore condition for the surface. This clarifies some previous work of F. H\'elein, Q. Xia-Y Shen, X. Ma and others, and, for instance, allows for the treatment of the Bj\"orling problem for Willmore surfaces in the presence of umbilics.