We give a description of the space of conformally immersed surfaces of a given compact Riemann surface as a finite dimensional submanifold of the Hilbert space of half-densities. The square of the position vector of this submanifold is the Willmore energy. The tangential projection of the position vector provides a vector field on the space of conformally immersed surfaces from the given Riemann surface. As a toy example we discuss these ideas on the space of planar closed curves with the elastic energy to obtain a flow preserving winding number and whose stationary points are elastica.

We will explain the integrable systems structure on the space of Willmore tori arising from the Novikov-Veselov hierarchy. This structure can be used to gain a picture of the moduli space of all (constrained) Willmore tori and to compute many examples of such surfaces.