The Morse index of Willmore spheres and its relation to the geometry of minimal surfaces

I will explain what the Willmore Morse Index of unbranched Willmore spheres in Euclidean three-space is and how to compute it. It turns out that several geometric properties at the ends of complete minimal surfaces with embedded planar ends are related to the mentioned Morse index.
One consequence of that computation is that all unbranched Willmore spheres are unstable (except for the round sphere). This talk is based on work with Jonas Hirsch.