In my talk I explain how Quaternionic Holomorphic Geometry can be used to discuss the associated family of isothermic surfaces. Isothermic surfaces are surfaces which have a conformal curvature line parametrisation, and cylinder, minimal surfaces and constant mean curvature surfaces are examples. It is known that any isothermic surface has a family of flat real connections such that parallel sections give new isothermic surfaces. In this talk, I will introduce a complex family which specialises to the known complex families for minimal surfaces and constant mean curvature surfaces.

Harmonic maps into appropriate spaces give rise to integrable systems; in particular, a spectral parameter can be introduced to investigate surfaces given by harmonicity. For example, surfaces with constant mean curvature have harmonic Gauss map. In the case of minimal surfaces the Gauss map does not determine the minimal surface uniquely but for surfaces with for non-vanishing mean curvature integrable system methods can been used to classify all CMC tori as meromorphic functions on the spectral curve. Recently, integrable systems have appeared in minimal surface theory, e.g., in the classification of planar domains or the classification of minimal annuli in S^2xR. To understand and exploit the link between classical tools of minimal surface theory and integrable system methods we propose to study instead of the Gauss map another harmonic map which includes the information of both the Gauss map and the support function of the minimal immersion. This is joint work with K. Moriya, Tsukuba.

There are various harmonic maps which are canonically associated to a minimal surface, e.g., the Gauss map of the immersion and the conformal Gauss map. In this talk, we will discuss how the well-known dressing operation applied to these harmonic maps is related to transformations on the minimal surface. In particular, we will show the link to the Lopez-Ros deformation, a generalised associated family and a family of Willmore surfaces given by the minimal surface. This is joint work with K. Moriya (University of Tsukuba).

Since the harmonicity of the conformal Gauss map characterizes Willmore surfaces, one can use harmonic map methods to obtain results for Willmore surfaces. In particular, we show that a Willmore torus with non-trivial normal bundle comes from holomorphic data by using an analogue of the delbar-sequence for harmonic maps into complex projective space. Moreover, the harmonic conformal Gauss map of a Willmore surface gives an associated family of flat connections, and thus allows to introduce a spectral parameter and to define a spectral curve. We will discuss both of these constructions, and the associated geometric transformations on Willmore surfaces.