Willmore energy of conformal immersions

Abstract

In this thesis, we summarize a result by Kuwert and Li [1] about the minimizers of the Willmore energy among the class of conformal immersions f : S^2 → R^3 with fixed isoperimetric ratio \sigma. Using results by Schygulla [2] as a starting point, we follow Kuwert and Li’s argument in [1] about the behavior of these minimizers as the isoperimetric ratio \sigma tends to zero. To do this, we first present some basic facts about conformal immersions, and then introduce some results by Chen and Li [3] about bubbles, concentration points, and the bubble tree. Then, we present the argument by Kuwert and Li [1] about the limiting behavior of the Willmore minimizers as the isoperimetric ratio \sigma goes to zero.