The Loewner Energy via the Renormalised Energy of Moving Frames

TL;DR

This paper presents a novel formula for the Loewner energy of Jordan curves on the sphere, linking it to the renormalised energy of moving frames and the Weil-Petersson Teichmüller space, revealing deep geometric connections.

Contribution

It introduces a new expression for the Loewner energy as a Kähler potential, connecting complex analysis, geometry, and Teichmüller theory in a novel way.

Findings

New formula for Loewner energy on the sphere

Connection between Loewner energy and Weil-Petersson metric

Identification of the energy as a Kähler potential

Abstract

We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a K\"ahler potential for the essentially unique K\"ahler metric on the Weil-Petersson universal Teichm\"uller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.