The analysis of constrained Willmore surfaces from the viewpoint of quaternionic function theory

Martin Schmidt (Mannheim)

Abstract

We use the quaternionic function theory of Pedit and Pinkall in order to describe constrained Willmore surfaces in the four and three dimensional sphere. We introduce a class of conformal immersions with square integrable Hopf fields and consider sequences in this class with bounded Willmore energy. The Willmore energy defines a bounded measure on the surface with a weak limit in the space of measures. As long as this limit does not contain point measures of weight al least \(4\pi\), the corresponding sequences have limits in the same class.