Informal seminar on geometry and integrable systems: Schmidt -- On constrained Willmore surfaces in R^4

Using quaternionic function theory, we present the Kodaira representation of conformal
maps of compact Riemann surfaces to R^4.  We reproduce the classical result, that the
subspace of conformal maps inside the space of immersions has a singularity at the
isothermic surfaces.  Our approach allows us to prove the existence of minimizers of the
Willmore functional without the assumption that the infimum of the Willmore functional
is less than 8 pi.  We present some generalisation of constrained Willmore tori, which
allows to deform their spectral curves to the unique vacuum with zero Willmore energy.
We want to use this to determine for any elliptic curve the minimum of the Willmore
energy on the space of conformal maps to R^4.