Abstract:

In a hyper-Kahler 4-manifold, holomorphic curves are stable minimal surfaces.  One may wonder whether those are all the stable minimal surfaces.  Micallef gave an affirmative answer in many cases.  However, this cannot be true in general.  The minimal sphere in the Atiyah--Hitchin manifold is a counter-example. In this talk, we will first recall the hyper-Kahler geometry of Atiyah--Hitchin manifold.  We will then explain that the minimal sphere is quite rigid in various senses.  This is based on a joint work with Mu-Tao Wang.