Gauss Maps of Conformal Surfaces in the Möbius n-sphere

In this note we discuss Gauss maps for conformal surfaces in the Möbius 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 surface and its dual. More generally, we define the concept of a Lorentzian 2-plane lift for an arbitrary conformal surface, and show that the conformal harmonicity of this lift is equivalent to the Willmore condition for the surface. Finally, S-Willmore surfaces are characterized by the primitive Lorentzian 2-plane lift. This clarifies some previous work of F. Hélein, Q. Xia–Y. Shen, X. Ma and others, and, for instance, allows for the treatment of the Björling problem for Willmore surfaces in the presence of umbilics.