Abstract: The conformal invariance of the Willmore functional has several consequences. One is that complete minimal surfaces in R3 are - after inversion at a sphere - Willmore surfaces. I will explain what the Morse Index (roughly speaking the number of direction that locally decrease the Willmore energy up to second order) of a compact Willmore sphere in R3 is and how this can be computed using the conformal invariance. It turns out that this Index is always positive except for the round sphere. This talk is based on work with Jonas Hirsch.
Willmore spheres are unstable