Complex Structure and Möbius Geometry

Abstract

This chapter takes up the Möbius invariant conformal structure on Möbius space. It induces a conformal structure on any immersed surface, which in turn induces a complex structure on the surface. Möbius geometry is the study of properties of conformal immersions of Riemann surfaces into Möbius space $\mathcal{M}$ that remain invariant under the action of $\mathbf{M\ddot{o}b}$. Each complex coordinate chart on an immersed surface has a unique Möbius frame field adapted to it, whose first order invariant we call k and whose second order invariant we call b. These are smooth, complex valued functions on the domain of the frame field. These frames are used to derive the structure equations for k and b, the conformal area, the conformal Gauss map, and the conformal area element. The equivariant embeddings of the space forms into Möbius space are conformal. Relative to a complex coordinate, the Hopf invariant, conformal factor, and mean curvature of an immersed surface in a space form determine the Möbius invariants k and b of the immersion into $\mathcal{M}$ obtained by applying the embedding of the space form into $\mathcal{M}$ to the given immersed surface. This gives a formula for the conformal area element showing that the Willmore energy is conformally invariant.