Remarks on the geometry of surfaces in the four-dimensional Mobius sphere

We study the conformal geometry of surfaces immersed in the fourdimensional conformal sphere Q4, viewed as a homogeneous space under the action of the M obius 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 vanishing results and some bounds on the Euler characteristic of the surfaces. Finally, we characterize isotropic surfaces through an Enneper-Weierstrass-type parametrization.