Remarkson the geometryofsurfacesinthe four-dimensional Möbius sphere

We study the conformal geometry of surfaces immersed in the four-dimensional conformal sphere Q4, viewed as a homogeneous space under the action of the Möbius group. We introduce the classes of ± isotropic surfaces and characterize them as those whose conformal Gauss map is antiholomorphic or holomorphic. We then relate these surfaces to Willmore surfaces and prove some interesting vanishing results and some bounds on the Euler characteristic of the surfaces. Finally, we characterize - isotropic surfaces through an Enneper-Weierstrass-typeparametrization.