Willmore Spheres in S^n via Loop Groups

Josef Dorfmeister (TU Muenchen)

This talk reports on ongoing work with Peng Wang (Tongji University).
We will consider Willmore surfaces in S^n via the loop group method.
For this we introduce a "Gauss map" which has the property that
an immersion is Willmore if and only if the Gauss map is
conformally harmonic.
Using a frame lift we will introduce a spectral parameter.
Specializing to Willmore surfaces from S^2 to S^n we show that the
Gauss map has finite uniton number. This allows to apply work of
Burstall and Guest. As a result we obtain normalized potentials
which are contained in some nilpotent Lie algebra. We will give
a fairly detailled description of these normalized potentials and
we will also discuss, how to construct all Willmore spheres in S^n.