The Loewner Energy via the Renormalised Energy of Moving Frames

Abstract

We obtain a new formula for the Loewner energy of simple curves of the sphere as the renormalised energy of moving frames of the two domains of the sphere delimited by the given curve. We also construct examples of surfaces of finite Willmore energy (or renormalised area) in H 3 bounding (a sub-class of) curves of finite Loewner energy using equipotential lines.