Minimal Surfaces of High Codimension: Isotropicity, Holomorphicity and Stability

The notions mentioned in the title are roughly related as follows. Isotropicity of a minimal surface is characterised by the vanishing of certain holomorphic differentials. Holomorphic curves in a complex torus with a flat metric are precisely the minimal surfaces which are maximally isotropic. And it is well known that a surface which is holomorphic in a Kahler manifold minimizes area in its homology class. I will present various results in this area and mention some open problems. In particular, I will discuss the deformation of a holomorphic curve in a complex torus with a flat metric to a minimal surface which is isotropic to a sufficiently high order (but less than maximal!). The holomorphicity of stable minimal surfaces which are isotropic to the same degree will also be described. This is joint work with Elisabeta Nedita and it is related to (some old) work with Claudio Arezzo and Jon Wolfson.