Existence of minimizing Willmore surfaces of prescribed conformal class

TL;DR

This paper proves the existence of smooth conformal mappings that minimize the Willmore functional within a class of mappings from compact Riemann surfaces into Euclidean space, extending quaternionic function theory to achieve this.

Contribution

It establishes the existence of minimizers for the Willmore functional in a conformal class and extends quaternionic function theory to square integrable Hopf fields.

Findings

Existence of smooth conformal minimizers for the Willmore functional.

Extension of quaternionic function theory to square integrable Hopf fields.

Proof of the Pluecker formula for these Hopf fields.

Abstract

We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed sequence converges. This implies that there exists a smooth conformal mapping, which minimizes the Willmore functional in this class. For this purpose we extend the quaternionic function theory of Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the Pluecker formula for such Hopf fields.