Conformal geometry in the four-dimensional Mobius space

In this thesis we study the geometry of surfaces immersed in the four-dimensional conformal sphere $Q_4$. It is known that a surface immersed in $Q_n$ is Willmore if and only if its conformal Gauss map is harmonic. Here we prove that a surface immersed in $Q_4$ is - or + isotropic if and only if its conformal Gauss map is, respectively, holomorphic or antiholomorphic. We make use of Cauchy-Riemann conditions to prove that, under suitable assumptions, a compact surface is either isotropic or its Euler characteristic is bounded from above. We then consider the notion of S-Willmore surface and prove that, even in the conformal setting, an S-Willmore surface is Willmore. We also prove that an isotropic surface is S-Willmore if and only if the curvature of the normal bundle associated with the surface does not vanish. Finally, we prove the existence of a bijection between the set of -isotropic, non totally umbilical, weakly conformal branched immersions of a fixed surface in $Q_4$, whose conformal gauss map can be continuously extended at the branch points, and non constant, holomorphic, totally isotropic maps with values in the conformal Grassmannian, with non constant associated map $J_\gamma$. The bijection is realized via the conformal gauss map